title: Fermat's Last Theorem

text: The special case , proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent "n" that is not a prime number, it must also be false for some smaller "n", so only prime values of "n" need further investigation. Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible (meaning that mathematicians generally considered a proof impossible, exceedingly difficult, or unachievable with current knowledge).


title: Fermat's Last Theorem

text: Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura conjecture (eventually as the modularity theorem), it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's theorem) widely considered completely inaccessible to proof.


title: Fermat's Last Theorem

text: In 1984, Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems. An outline suggesting this could be proved was given by Frey. The full proof that the two problems were closely linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture" (see: "Ribet's Theorem" and "Frey curve"). These papers by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true, then by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well.


title: Fermat's Last Theorem

text: Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Unlike Fermat's Last Theorem, the Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates' quoted reaction was a common one:


title: Fermat's Last Theorem

text: On hearing that Ribet had proven Frey's link to be correct, English mathematician Andrew Wiles, who had a childhood fascination with Fermat's Last Theorem and had a background of working with elliptic curves and related fields, decided to try to prove the Taniyama–Shimura conjecture as a way to prove Fermat's Last Theorem. In 1993, after six years of working secretly on the problem, Wiles succeeded in proving enough of the conjecture to prove Fermat's Last Theorem. Wiles's paper was massive in size and scope. A flaw was discovered in one part of his original paper during peer review and required a further year and collaboration with a past student, Richard Taylor, to resolve. As a result, the final proof in 1995 was accompanied by a smaller joint paper showing that the fixed steps were valid. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the Taniyama–Shimura–Weil conjecture, now proven and known as the modularity theorem, were subsequently proved by other mathematicians, who built on Wiles's work between 1996 and 2001. For his proof, Wiles was honoured and received numerous awards, including the 2016 Abel Prize.


title: Fermat's Last Theorem

text: There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem.


title: Fermat's Last Theorem

text: In order to state them, we use mathematical notation: let be the set of natural numbers 1, 2, 3, ..., let be the set of integers 0, ±1, ±2, ..., and let be the set of rational numbers , where and are in with . In what follows we will call a solution to where one or more of , , or is zero a "trivial solution". A solution where all three are non-zero will be called a "non-trivial" solution.


title: Fermat's Last Theorem

text: The equivalence is clear if is even. If is odd and all three of are negative, then we can replace with to obtain a solution in . If two of them are negative, it must be and or and . If are negative and is positive, then we can rearrange to get resulting in a solution in ; the other case is dealt with analogously. Now if just one is negative, it must be or . If is negative, and and are positive, then it can be rearranged to get again resulting in a solution in ; if is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in would also mean a solution exists in , the original formulation of the problem.


title: Fermat's Last Theorem

text: This is because the exponents of and are equal (to ), so if there is a solution in , then it can be multiplied through by an appropriate common denominator to get a solution in , and hence in .


title: Fermat's Last Theorem

text: A non-trivial solution , , ∈ to yields the non-trivial solution , ∈ for . Conversely, a solution , ∈ to yields the non-trivial solution for .


title: Fermat's Last Theorem

text: This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over the field , rather than over the ring ; fields exhibit more structure than rings, which allows for deeper analysis of their elements.


title: Fermat's Last Theorem

text: Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. However, the proof by Andrew Wiles proves that any equation of the form does have a modular form. Any non-trivial solution to (with an odd prime) would therefore create a contradiction, which in turn proves that no non-trivial solutions exist.


title: Fermat's Last Theorem

text: In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be "attacked" for all numbers at once.


title: Fermat's Last Theorem

text: In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together , equals the square of the length of the third side , would also be a right angle triangle.


title: Fermat's Last Theorem

text: This is now known as the Pythagorean theorem, and a triple of numbers that meets this condition is called a Pythagorean triple – both are named after the ancient Greek Pythagoras. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, and methods for generating such triples have been studied in many cultures, beginning with the Babylonians and later ancient Greek, Chinese, and Indian mathematicians. Mathematically, the definition of a Pythagorean triple is a set of three integers ("a", "b", "c") that satisfy the equation formula_1


title: Fermat's Last Theorem

text: Fermat's equation, "x" + "y" = "z" with positive integer solutions, is an example of a Diophantine equation, named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers "x" and "y" such that their sum, and the sum of their squares, equal two given numbers "A" and "B", respectively:


title: Fermat's Last Theorem

text: Diophantus's major work is the "Arithmetica", of which only a portion has survived. Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the "Arithmetica", that was translated into Latin and published in 1621 by Claude Bachet.


title: Fermat's Last Theorem

text: Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation "x" + "y" = "z" are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26"x" + 65"y" = 13, may be found using the Euclidean algorithm (c. 5th century BC).


title: Fermat's Last Theorem

text: Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no "cross terms" mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers "x", "y", and "z" such that "x" + "y" = "z" where "n" and "m" are relatively prime natural numbers.


title: Fermat's Last Theorem

text: Problem II.8 of the "Arithmetica" asks how a given square number is split into two other squares; in other words, for a given rational number "k", find rational numbers "u" and "v" such that "k" = "u" + "v". Diophantus shows how to solve this sum-of-squares problem for "k" = 4 (the solutions being "u" = 16/5 and "v" = 12/5).


title: Fermat's Last Theorem

text: Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the "Arithmetica" next to Diophantus's sum-of-squares problem:


title: Fermat's Last Theorem

text: After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments. Although not actually a theorem at the time (meaning a mathematical statement for which proof exists), the marginal note became known over time as "Fermat’s Last Theorem", as it was the last of Fermat's asserted theorems to remain unproved.


title: Fermat's Last Theorem

text: It is not known whether Fermat had actually found a valid proof for all exponents "n", but it appears unlikely. Only one related proof by him has survived, namely for the case "n" = 4, as described in the section "Proofs for specific exponents".


title: Fermat's Last Theorem

text: While Fermat posed the cases of "n" = 4 and of "n" = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis, he never posed the general case. Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes Weil as saying Fermat must have briefly deluded himself with an irretrievable idea.


title: Fermat's Last Theorem

text: Wiles and Taylor's proof relies on 20th-century techniques. Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time.


title: Fermat's Last Theorem

text: While Harvey Friedman's grand conjecture implies that any provable theorem (including Fermat's last theorem) can be proved using only 'elementary function arithmetic', such a proof need be 'elementary' only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat's proof.


title: Fermat's Last Theorem

text: Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer. His proof is equivalent to demonstrating that the equation


title: Fermat's Last Theorem

text: has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem for the case "n" = 4, since the equation "a" + "b" = "c" can be written as "c" − "b" = ("a").


title: Fermat's Last Theorem

text: Alternative proofs of the case "n" = 4 were developed later by Frénicle de Bessy (1676), Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830), Schopis (1825), Olry Terquem (1846), Joseph Bertrand (1851), Victor Lebesgue (1853, 1859, 1862), Théophile Pépin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901), Gambioli (1901), Leopold Kronecker (1901), Bang (1905), Sommer (1907), Bottari (1908), Karel Rychlík (1910), Nutzhorn (1912), Robert Carmichael (1913), Hancock (1931), Gheorghe Vrănceanu (1966), Grant and Perella (1999), Barbara (2007), and Dolan (2011).


title: Fermat's Last Theorem

text: After Fermat proved the special case "n" = 4, the general proof for all "n" required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation "a" + "b" = "c" has no positive integer solutions ("a", "b", "c") when "n" is an odd prime number. This follows because a solution ("a", "b", "c") for a given "n" is equivalent to a solution for all the factors of "n". For illustration, let "n" be factored into "d" and "e", "n" = "de". The general equation


title: Fermat's Last Theorem

text: Thus, to prove that Fermat's equation has no solutions for "n" > 2, it would suffice to prove that it has no solutions for at least one prime factor of every "n". Each integer "n" > 2 is divisible by 4 or by an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all "n" if it could be proved for "n" = 4 and for all odd primes "p".


title: Fermat's Last Theorem

text: In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents "p" = 3, 5 and 7. The case "p" = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect. In 1770, Leonhard Euler gave a proof of "p" = 3, but his proof by infinite descent contained a major gap. However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof. Independent proofs were published by Kausler (1802), Legendre (1823, 1830), Calzolari (1855), Gabriel Lamé (1865), Peter Guthrie Tait (1872), Günther (1878), Gambioli (1901), Krey (1909), Rychlík (1910), Stockhaus (1910), Carmichael (1915), Johannes van der Corput (1915), Axel Thue (1917), and Duarte (1944).


title: Fermat's Last Theorem

text: The case "p" = 5 was proved independently by Legendre and Peter Gustav Lejeune Dirichlet around 1825. Alternative proofs were developed by Carl Friedrich Gauss (1875, posthumous), Lebesgue (1843), Lamé (1847), Gambioli (1901), Werebrusow (1905), Rychlík (1910), van der Corput (1915), and Guy Terjanian (1987).


title: Fermat's Last Theorem

text: The case "p" = 7 was proved by Lamé in 1839. His rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs were published by Angelo Genocchi in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin (1876) and Edmond Maillet (1897).


title: Fermat's Last Theorem

text: Fermat's Last Theorem was also proved for the exponents "n" = 6, 10, and 14. Proofs for "n" = 6 were published by Kausler, Thue, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly, Dirichlet and Terjanian each proved the case "n" = 14, while Kapferer and Breusch each proved the case "n" = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for "n" = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for "n" = 14 was published in 1832, before Lamé's 1839 proof for "n" = 7.


title: Fermat's Last Theorem

text: All proofs for specific exponents used Fermat's technique of infinite descent, either in its original form, or in the form of descent on elliptic curves or abelian varieties. The details and auxiliary arguments, however, were often "ad hoc" and tied to the individual exponent under consideration. Since they became ever more complicated as "p" increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents. Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow, the first significant work on the general theorem was done by Sophie Germain.


title: Fermat's Last Theorem

text: In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents. First, she defined a set of auxiliary primes formula_5 constructed from the prime exponent formula_6 by the equation formula_7, where formula_8 is any integer not divisible by three. She showed that, if no integers raised to the formula_9 power were adjacent modulo formula_5 (the "non-consecutivity condition"), then formula_5 must divide the product formula_12. Her goal was to use mathematical induction to prove that, for any given formula_6, infinitely many auxiliary primes formula_5 satisfied the non-consecutivity condition and thus divided formula_12; since the product formula_12 can have at most a finite number of prime factors, such a proof would have established Fermat's Last Theorem. Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. She also worked to set lower limits on the size of solutions to Fermat's equation for a given exponent formula_6, a modified version of which was published by Adrien-Marie Legendre. As a byproduct of this latter work, she proved Sophie Germain's theorem, which verified the first case of Fermat's Last Theorem (namely, the case in which formula_6 does not divide formula_12) for every odd prime exponent less than formula_20, and for all primes formula_6 such that at least one of formula_22, formula_23, formula_24, formula_25, formula_26 and formula_27 is prime (specially, the primes formula_6 such that formula_22 is prime are called Sophie Germain primes). Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for formula_30, which was proved by Guy Terjanian in 1977. In 1985, Leonard Adleman, Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes formula_6.


title: Fermat's Last Theorem

text: In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation in complex numbers, specifically the cyclotomic field based on the roots of the number 1. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers. This gap was pointed out immediately by Joseph Liouville, who later read a paper that demonstrated this failure of unique factorisation, written by Ernst Kummer.


title: Fermat's Last Theorem

text: Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the ideal numbers.


title: Fermat's Last Theorem

text: Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) that conjecturally occur approximately 39% of the time; the only irregular primes below 270 are 37, 59, 67, 101, 103, 131, 149, 157, 233, 257 and 263.


title: Fermat's Last Theorem

text: In the 1920s, Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent "n" is greater than two. This conjecture was proved in 1983 by Gerd Faltings, and is now known as Faltings's theorem.


title: Fermat's Last Theorem

text: In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. In 1954, Harry Vandiver used a SWAC computer to prove Fermat's Last Theorem for all primes up to 2521. By 1978, Samuel Wagstaff had extended this to all primes less than 125,000. By 1993, Fermat's Last Theorem had been proved for all primes less than four million.


title: Fermat's Last Theorem

text: However, despite these efforts and their results, no proof existed of Fermat's Last Theorem. Proofs of individual exponents by their nature could never prove the "general" case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. (This had been the case with some other past conjectures, and it could not be ruled out in this conjecture.)


title: Fermat's Last Theorem

text: The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding" Taniyama–Shimura–Weil conjecture, proposed around 1955—which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now known as the modularity theorem.


title: Fermat's Last Theorem

text: Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.


title: Fermat's Last Theorem

text: The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture.


title: Fermat's Last Theorem

text: Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof. For example, Wiles's doctoral supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."


title: Fermat's Last Theorem

text: In 1984, Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution ("a", "b", "c") for exponent "p" > 2, then it could be shown that the semi-stable elliptic curve (now known as a Frey-Hellegouarch)


title: Fermat's Last Theorem

text: would have such unusual properties that it was unlikely to be modular. This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. As such, Frey observed that a proof of the Taniyama–Shimura–Weil conjecture might also simultaneously prove Fermat's Last Theorem. By contraposition, a "disproof" or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture.


title: Fermat's Last Theorem

text: In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 4 numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the latter were true, the former could not be disproven, and would also have to be true.


title: Fermat's Last Theorem

text: Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem – or at least to prove it for the types of elliptical curves that included Frey's equation (known as semistable elliptic curves). This was widely believed inaccessible to proof by contemporary mathematicians. Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey showed that this was "plausible" but did not go as far as giving a full proof. The missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was identified by Jean-Pierre Serre who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by Ken Ribet.


title: Fermat's Last Theorem

text: Ribet's proof of the epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, Andrew Wiles, an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves.


title: Fermat's Last Theorem

text: Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. His initial study suggested proof by induction, and he based his initial work and first significant breakthrough on Galois theory before switching to an attempt to extend horizontal Iwasawa theory for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem. However, by mid-1991, Iwasawa theory also seemed to not be reaching the central issues in the problem. In response, he approached colleagues to seek out any hints of cutting-edge research and new techniques, and discovered an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof. Wiles studied and extended this approach, which worked. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 1993 he asked his Princeton colleague, Nick Katz, to help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.


title: Fermat's Last Theorem

text: By mid-May 1993, Wiles was ready to tell his wife he thought he had solved the proof of Fermat's Last Theorem, and by June he felt sufficiently confident to present his results in three lectures delivered on 21–23 June 1993 at the Isaac Newton Institute for Mathematical Sciences. Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it became apparent during peer review that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular group. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer), who alerted Wiles on 23 August 1993.


title: Fermat's Last Theorem

text: The error would not have rendered his work worthless – each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected. However without this part proved, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor, without success. By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether it was complete or not, so that the wider community could explore and use whatever he had managed to accomplish. But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.


title: Fermat's Last Theorem

text: Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and fix the error. He adds that he was having a final look to try and understand the fundamental reasons for why his approach could not be made to work, when he had a sudden insight – that the specific reason why the Kolyvagin–Flach approach would not work directly "also" meant that his original attempts using Iwasawa theory could be made to work, if he strengthened it using his experience gained from the Kolyvagin–Flach approach. Fixing one approach with tools from the other approach would resolve the issue for all the cases that were not already proven by his refereed paper. He described later that Iwasawa theory and the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle.


title: Fermat's Last Theorem

text: On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the "Annals of Mathematics". The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an "R=T theorem") to prove modularity lifting theorems has been an influential development in algebraic number theory.


title: Fermat's Last Theorem

text: These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.


title: Fermat's Last Theorem

text: The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad et al. (1999), and Breuil et al. (2001) who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.


title: Fermat's Last Theorem

text: Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime "n"-th powers, "n" ≥ 3. (The case "n" = 3 was already known by Euler.)


title: Fermat's Last Theorem

text: Fermat's Last Theorem considers solutions to the Fermat equation: with positive integers , , and and an integer greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent to be a negative integer or rational, or to consider three different exponents.


title: Fermat's Last Theorem

text: The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions "a, b, c, m, n, k" satisfying


title: Fermat's Last Theorem

text: In particular, the exponents "m", "n", "k" need not be equal, whereas Fermat's last theorem considers the case


title: Fermat's Last Theorem

text: The Beal conjecture, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture, states that there are no solutions to the generalized Fermat equation in positive integers "a", "b", "c", "m", "n", "k" with "a", "b", and "c" being pairwise coprime and all of "m", "n", "k" being greater than 2.


title: Fermat's Last Theorem

text: The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. The conjecture states that the generalized Fermat equation has only "finitely many" solutions ("a", "b", "c", "m", "n", "k") with distinct triplets of values ("a", "b", "c"), where "a", "b", "c" are positive coprime integers and "m", "n", "k" are positive integers satisfying


title: Fermat's Last Theorem

text: When we allow the exponent to be the reciprocal of an integer, i.e. for some integer , we have the inverse Fermat equation


title: Fermat's Last Theorem

text: All solutions of this equation were computed by Hendrik Lenstra in 1992. In the case in which the "m" roots are required to be real and positive, all solutions are given by


title: Fermat's Last Theorem

text: For the Diophantine equation formula_36 with "n" not equal to 1, Bennett, Glass, and Székely proved in 2004 for "n" > 2, that if "n" and "m" are coprime, then there are integer solutions if and only if 6 divides "m", and formula_37, formula_38 and formula_39 are different complex 6th roots of the same real number.


title: Fermat's Last Theorem

text: All primitive integer solutions (i.e., those with no prime factor common to all of "a", "b", and "c") to the optic equation formula_40 can be written as


title: Fermat's Last Theorem

text: The case "n" = −2 also has an infinitude of solutions, and these have a geometric interpretation in terms of right triangles with integer sides and an integer altitude to the hypotenuse. All primitive solutions to formula_44 are given by


title: Fermat's Last Theorem

text: for coprime integers "u", "v" with "v" > "u". The geometric interpretation is that "a" and "b" are the integer legs of a right triangle and "d" is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer


title: Fermat's Last Theorem

text: There are no solutions in integers for formula_49 for integers "n" < −2. If there were, the equation could be multiplied through by formula_50 to obtain formula_51, which is impossible by Fermat's Last Theorem.


title: Fermat's Last Theorem

text: The abc conjecture roughly states that if three positive integers "a", "b" and "c" (hence the name) are coprime and satisfy "a" + "b" = "c", then the radical "d" of "abc" is usually not much smaller than "c". In particular, the abc conjecture in its most standard formulation implies Fermat's last theorem for "n" that are sufficiently large. The modified Szpiro conjecture is equivalent to the abc conjecture and therefore has the same implication. An effective version of the abc conjecture, or an effective version of the modified Szpiro conjecture, implies Fermat's Last Theorem outright.


title: Fermat's Last Theorem

text: In 1816, and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. In 1857, the Academy awarded 3,000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize. Another prize was offered in 1883 by the Academy of Brussels.


title: Fermat's Last Theorem

text: In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks—a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a complete proof of Fermat's Last Theorem. On 27 June 1908, the Academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun. Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997. In March 2016, Wiles was awarded the Norwegian government's Abel prize worth €600,000 for "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory."


title: Fermat's Last Theorem

text: Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3 meters) of correspondence. In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to some claims, Edmund Landau tended to use a special preprinted form for such proofs, where the location of the first mistake was left blank to be filled by one of his graduate students. According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career". In the words of mathematical historian Howard Eves, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."


title: Fermat's Last Theorem

text: The popularity of the theorem outside science has led to it being described as achieving "that rarest of mathematical accolades: A niche role in pop culture."


title: Fermat's Last Theorem

text: Arthur Porges' 1954 short story "The Devil and Simon Flagg" features a mathematician who bargains with the Devil that the latter cannot produce a proof of Fermat's Last Theorem within twenty-four hours.


title: Fermat's Last Theorem

text: In "The Simpsons" episode "The Wizard of Evergreen Terrace," Homer Simpson writes the equation formula_52 on a blackboard, which appears to be a counterexample to Fermat's Last Theorem. The equation is wrong, but it appears to be correct if entered in a calculator with 10 significant figures.


title: Popcorn

text: Popcorn (also called popped corn, popcorns or pop-corn) is a variety of corn kernel which expands and puffs up when heated; the same names also refer to the foodstuff produced by the expansion.


title: Popcorn

text: A popcorn kernel's strong hull contains the seed's hard, starchy shell endosperm with 14–20% moisture, which turns to steam as the kernel is heated. Pressure from the steam continues to build until the hull ruptures, allowing the kernel to forcefully expand, to 20 to 50 times its original size, and then cool.


title: Popcorn

text: Some strains of corn (taxonomized as "Zea mays") are cultivated specifically as popping corns. The "Zea mays" variety "everta", a special kind of flint corn, is the most common of these.


title: Popcorn

text: Popcorn is one of six major types of corn, which includes dent corn, flint corn, pod corn, flour corn, and sweet corn.


title: Popcorn

text: Corn was domesticated about 10,000 years ago, in what is now Mexico. Archaeologists discovered that people have known about popcorn for thousands of years. Fossil evidence from Peru suggests that corn was popped as early as 4,700 BC.


title: Popcorn

text: Through the 19th century, popping of the kernels was achieved by hand, on stove tops. Kernels were sold on the East Coast of the United States under names such as "Pearls" or "Nonpareil". The term "popped corn" first appeared in John Russell Bartlett's 1848 "Dictionary of Americanisms". Popcorn is an ingredient in Cracker Jack and, in the early years of the product, it was popped by hand.


title: Popcorn

text: Popcorn's accessibility increased rapidly in the 1890s with Charles Cretors' invention of the popcorn maker. Cretors, a Chicago candy store owner, had created a number of steam-powered machines for roasting nuts and applied the technology to the corn kernels.


title: Popcorn

text: By the turn of the century, Cretors had created and deployed street carts equipped with steam-powered popcorn makers.During the Great Depression, popcorn was fairly inexpensive at 5–10 cents a bag and became popular. Thus, while other businesses failed, the popcorn business thrived and became a source of income for many struggling farmers, including the Redenbacher family, namesake of the famous popcorn brand. During World War II, sugar rations diminished candy production, and Americans compensated by eating three times as much popcorn as they had before. The snack was popular at theaters, much to the initial displeasure of many of the theater owners, who thought it distracted from the films. Their minds eventually changed, however, and in 1938 a Midwestern theater owner named Glen W. Dickinson Sr. installed popcorn machines in the lobbies of his Dickinson theaters. Popcorn was more profitable than theater tickets, and at the suggestion of his production consultant, R. Ray Aden, Dickinson purchased popcorn farms and was able to keep ticket prices down. The venture was a success, and popcorn soon spread. The rise of television in the 1940s brought lower popcorn consumption as theater attendance fell. The Popcorn Institute (a trade association of popcorn processors) promoted popcorn consumption at home, bringing it back to previous levels.


title: Popcorn

text: At least six localities (all in the Midwestern United States) claim to be the "Popcorn Capital of the World;": Ridgway, Illinois; Valparaiso, Indiana; Van Buren, Indiana; Schaller, Iowa; Marion, Ohio; and North Loup, Nebraska. According to the USDA, specific corn for popcorn is grown, mostly in Nebraska and Indiana; increasingly in Texas. As the result of an elementary school project, popcorn became the official state snack food of Illinois.


title: Popcorn

text: Each kernel of popcorn contains moisture and oil. Unlike most other grains, the outer hull of the popcorn kernel is strong and impervious to moisture, and the starch inside consists almost entirely of a hard type.


title: Popcorn

text: As the oil and water in the kernel are heated, they turn into steam. Under these conditions, the starch inside the kernel gelatinizes and softens. The steam pressure increases until the breaking point of the hull is reached; a pressure of approximately and a temperature of . The hull ruptures, causing a sudden drop in pressure inside the kernel and a corresponding rapid expansion of the steam, which expands the starch and proteins of the endosperm into airy foam. As the foam rapidly cools, the starch and protein polymers set into the familiar crispy puff. Special varieties are grown to improve popping yield. Though the kernels of some other types will pop, the cultivated strain for popcorn is "Zea mays everta," which is a variety of flint corn.


title: Popcorn

text: Popcorn can be cooked with butter or oil. Although small quantities can be popped in a stove-top kettle or pot in a home kitchen, commercial sale employs specially designed popcorn machines, which were invented in Chicago, Illinois, by Charles Cretors in 1885. Cretors introduced his invention at the Columbian Exposition in 1893. At that fair, F. W. Rueckheim introduced a molasses-flavored "Candied Popcorn", the first caramel corn; his brother, Louis Ruekheim, slightly altered the recipe and introduced it as Cracker Jack in 1896.


title: Popcorn

text: Cretors's invention was the first patented steam-driven machine that popped corn in oil. Previously, vendors popped corn by holding a wire basket over an open flame. At best, the result was hot, dry, and unevenly cooked. Cretors's machine popped corn in a mixture of one-third clarified butter, two-thirds lard, and salt. This mixture can withstand the temperature needed to pop corn and produces little smoke. A fire under a boiler created steam that drove a small engine to drive gears, shaft, and the agitator that stirred the corn, and also powered a small puppet, "The Toasty Roasty Man", an attention-getting amusement to attract business. A wire connected to the top of the cooking pan allowed the operator to disengage the drive mechanism, lift the cover, and dump popped corn into the storage bin beneath. Exhaust from the steam engine was piped to a pan below the corn storage bin and kept freshly popped corn warm. Excess steam was also used to operate a small, shrill whistle to attract attention.


title: Popcorn

text: A different method of popcorn-making involves the "popcorn hammer", a large cast-iron canister that is sealed with a heavy lid and slowly turned over a fire in rotisserie fashion.


title: Popcorn

text: Popping results are sensitive to the rate at which the kernels are heated. If heated too quickly, the steam in the outer layers of the kernel can reach high pressures and rupture the hull before the starch in the center of the kernel can fully gelatinize, leading to partially popped kernels with hard centers. Heating too slowly leads to entirely unpopped kernels: the tip of the kernel, where it attached to the cob, is not entirely moisture-proof, and when heated slowly, the steam can leak out of the tip fast enough to keep the pressure from rising sufficiently to break the hull and cause the pop.


title: Popcorn

text: Producers and sellers of popcorn consider two major factors in evaluating the quality of popcorn: what percentage of the kernels will pop, and how much each popped kernel expands. Expansion is an important factor to both the consumer and vendor. For the consumer, larger pieces of popcorn tend to be more tender and are associated with higher quality. For the grower, distributor and vendor, expansion is closely correlated with profit: vendors such as theaters buy popcorn by weight and sell it by volume. For these reasons, higher-expansion popcorn fetches a higher profit per unit weight.


title: Popcorn

text: Popcorn will pop when freshly harvested, but not well; its high moisture content leads to poor expansion and chewy pieces of popcorn. Kernels with a high moisture content are also susceptible to mold when stored. For these reasons, popcorn growers and distributors dry the kernels until they reach the moisture level at which they expand the most. This differs by variety and conditions, but is generally in the range of 14–15% moisture by weight. If the kernels are over-dried, the expansion rate will suffer and the percentage of kernels that pop will decline. Old popcorn tends to dry out, lowering the yield.


title: Popcorn

text: When the popcorn has finished popping, sometimes unpopped kernels remain. Known in the popcorn industry as "old maids", these kernels fail to pop because they do not have enough moisture to create enough steam for an explosion. Re-hydrating prior to popping usually results in eliminating the unpopped kernels.


title: Popcorn

text: Popcorn varieties are broadly categorized by the shape of the kernels, the color of the kernels, or the shape of the popped corn. While the kernels may come in a variety of colors, the popped corn is always off-yellow or white as it is only the hull (or pericarp) that is colored. "Rice" type popcorn have a long kernel pointed at both ends; "pearl" type kernels are rounded at the top. Commercial popcorn production has moved mostly to pearl types. Historically, pearl popcorn were usually yellow and rice popcorn usually white. Today both shapes are available in both colors, as well as others including black, red, mauve, purple, and variegated. Mauve and purple popcorn usually have smaller and nutty kernels. Commercial production is dominated by white and yellow.


title: Popcorn

text: In the popcorn industry, a popped kernel of corn is known as a "flake". Two shapes of flakes are commercially important. "Butterfly" (or "snowflake") flakes are irregular in shape and have a number of protruding "wings". "Mushroom" flakes are largely ball-shaped, with few wings. Butterfly flakes are regarded as having better mouthfeel, with greater tenderness and less noticeable hulls. Mushroom flakes are less fragile than butterfly flakes and are therefore often used for packaged popcorn or confectionery, such as caramel corn. The kernels from a single cob of popcorn may form both butterfly and mushroom flakes; hybrids that produce 100% butterfly flakes or 100% mushroom flakes exist, the latter developed only as recently as 1998. Growing conditions and popping environment can also affect the butterfly-to-mushroom ratio.


title: Popcorn

text: When referring to multiple pieces of popcorn collectively, it is acceptable to use the term "popcorn". When referring to a singular piece of popcorn, the accepted term is "kernel".


title: Popcorn

text: Popcorn is a popular snack food at sporting events and in movie theaters, where it has been served since the 1930s. Cinemas have come under fire due to their high markup on popcorn; Stuart Hanson, a film historian at De Montfort University in Leicester, once said, "One of the great jokes in the industry is that popcorn is second only to cocaine or heroin in terms of profit."


title: Popcorn

text: Traditions differ as to whether popcorn is consumed as a hearty snack food with salt (predominating in the United States) or as a sweet snack food with caramelized sugar (predominating in Germany).


title: Popcorn

text: Popcorn smell has an unusually attractive quality for human beings. This is largely because it contains high levels of the chemicals 6-acetyl-2,3,4,5-tetrahydropyridine and 2-acetyl-1-pyrroline, very powerful aroma compounds that are also used by food and other industries either to make products that smell like popcorn, bread, or other foods containing the compound in nature, or for other purposes.


title: Popcorn

text: Popcorn as a breakfast cereal was consumed by Americans in the 1800s and generally consisted of popcorn with milk and a sweetener.


title: Popcorn

text: Popcorn balls (popped kernels stuck together with a sugary "glue") were hugely popular around the turn of the 20th century, but their popularity has since waned. Popcorn balls are still served in some places as a traditional Halloween treat. Cracker Jack is a popular, commercially produced candy that consists of peanuts mixed in with caramel-covered popcorn. Kettle corn is a variation of normal popcorn, cooked with white sugar and salt, traditionally in a large copper kettle. Once reserved for specialty shops and county fairs, kettle corn has recently become popular, especially in the microwave popcorn market. The popcorn maker is a relatively new home appliance, and its popularity is increasing because it offers the opportunity to add flavors of the consumer's own choice and to choose healthy-eating popcorn styles.


title: Popcorn

text: Popped sorghum is popular as a snack in India. The popped sorghum is similar to popcorn, but the puffs are smaller. Recipes for popping sorghum by microwave, in a pot, etc, are readily available online.


title: Popcorn

text: Air-popped popcorn (no salt or other additives) is 4% water, 78% carbohydrates (including 15% dietary fiber), 12% protein, and 4% fat (table). In a 100 gram reference amount, popcorn provides 382 calories and is a rich source (20% or more of the Daily Value, DV) of riboflavin (25% DV) and several dietary minerals, particularly manganese, phosphorus, and zinc (36-45% DV). B vitamins and other minerals are in appreciable amounts (table).


title: Popcorn

text: Movie theaters commonly use coconut oil to pop the corn, and then top it with butter or margarine. Movie theater popcorn contains large amounts of saturated fats and sodium due to it's method of preparation.


title: Popcorn

text: Popcorn is included on the list of foods that the American Academy of Pediatrics recommends not serving to children under four, because of the risk of choking.


title: Popcorn

text: Microwaveable popcorn represents a special case, since it is designed to be cooked along with its various flavoring agents. One of these formerly common artificial-butter flavorants, diacetyl, has been implicated in causing respiratory illnesses in microwave popcorn factory workers, also known as "popcorn lung". Major manufacturers in the United States have stopped using this chemical, including Orville Redenbacher's, Act II, Pop Secret and Jolly Time.


title: Popcorn

text: Popcorn, threaded onto a string, is used as a wall or Christmas tree decoration in some parts of North America, as well as on the Balkan peninsula.


title: Popcorn

text: Some shipping companies have experimented with using popcorn as a biodegradable replacement for expanded polystyrene packing material. However, popcorn has numerous undesirable properties as a packing material, including attractiveness to pests, flammability, and a higher cost and greater density than expanded polystyrene. A more processed form of expanded corn foam has been developed to overcome some of these limitations, forming starch-based foam peanuts.


title: Black Friday (shopping)

text: Black Friday is a colloquial term for the Friday after Thanksgiving in the United States. It traditionally marks the start of the Christmas shopping season in the United States. Many stores


title: Black Friday (shopping)

text: offer highly promoted sales at discounted prices and often open early, sometimes as early as midnight or even on Thanksgiving. Some stores' sales continue to Monday ("Cyber Monday") or for a week ("Cyber Week").


title: Black Friday (shopping)

text: Black Friday occurs on the day after Thanksgiving in the United States. Black Friday has routinely been the busiest shopping day of the year in the United States.


title: Black Friday (shopping)

text: For centuries, the adjective "black" has been applied to days upon which calamities occurred. Many events have been described as "Black Friday", although the most significant such event in American history was the Panic of 1869, which occurred when financiers Jay Gould and James Fisk took advantage of their connections with the Grant Administration in an attempt to corner the gold market. When President Grant learned of this manipulation, he ordered the Treasury to release a large supply of gold, which halted the run and caused prices to drop by 18%. Fortunes were made and lost in a single day, and the president's own brother-in-law, Abel Corbin, was ruined.


title: Black Friday (shopping)

text: The earliest known use of "Black Friday" to refer to the day after Thanksgiving occurred in the journal, "Factory Management and Maintenance", for November 1951, and again in 1952. Here it referred to the practice of workers calling in sick on the day after Thanksgiving, in order to have a four-day week-end. However, this use does not appear to have caught on. Around the same time, the terms "Black Friday" and "Black Saturday" came to be used by the police in Philadelphia and Rochester to describe the crowds and traffic congestion accompanying the start of the Christmas shopping season. In 1961, the city and merchants of Philadelphia attempted to improve conditions, and a public relations expert recommended rebranding the days "Big Friday" and "Big Saturday,” but these terms were quickly forgotten.


title: Black Friday (shopping)

text: The use of the phrase spread slowly, first appearing in "The New York Times" on November 29, 1975, in which it still refers specifically to "the busiest shopping and traffic day of the year" in Philadelphia. Although it soon became more widespread, "The Philadelphia Inquirer" reported in 1985 that retailers in Cincinnati and Los Angeles were still unaware of the term.


title: Black Friday (shopping)

text: As the phrase gained national attention in the early 1980s, merchants objecting to the use of a derisive term to refer to one of the most important shopping days of the year suggested an alternative derivation: that retailers traditionally operated at a financial loss for most of the year (January through November) and made their profit during the holiday season, beginning on the day after Thanksgiving. When this was recorded in the financial records, once-common accounting practices would use red ink to show negative amounts and black ink to show positive amounts. Black Friday, under this theory, is the beginning of the period when retailers would no longer be "in the red,” instead of taking in the year's profits. The earliest known published reference to this explanation occurs in "The Philadelphia Inquirer" for November 28, 1981.


title: Black Friday (shopping)

text: Since the early 21st century, there have been attempts by U.S.-based retailers to introduce a retail "Black Friday" to other countries around the world. Retailers outside the US have attempted to promote the day to remain competitive with US-based online retailers.


title: Black Friday (shopping)

text: In more recent decades, global retailers have adopted the term and date to market their own holiday sales.


title: Black Friday (shopping)

text: The day after Thanksgiving has been regarded as the beginning of the United States Christmas shopping season since 1952. The practice may be linked with the idea of Santa Claus parades. Parades celebrating Thanksgiving often include an appearance by Santa at the end of the parade, with the idea that "Santa has arrived" or "Santa is just around the corner" because Christmas is always the next major Christian holiday following Thanksgiving.


title: Black Friday (shopping)

text: In the late 19th and early 20th centuries, many Santa or Thanksgiving parades were sponsored by department stores. These include the continuing Toronto Santa Claus Parade in Canada since 1905, originally sponsored by Eaton's, and Macy's Thanksgiving Day Parade held in Manhattan, New York City since 1924, sponsored by Macy's. Department stores would use the parades to launch a big advertising push. Eventually, it became an unwritten rule that no store would try doing Christmas advertising before the parade was over. Therefore, the day after Thanksgiving became the day when the shopping season officially started.


title: Black Friday (shopping)

text: Thanksgiving's relationship to Christmas shopping led to controversy in the 1930s. Retail stores would have liked to have a longer shopping season, but no store wanted to break with tradition and be the one to start advertising before Thanksgiving. For this reason, in 1939, President Franklin D. Roosevelt issued a presidential proclamation proclaiming Thanksgiving to be the fourth Thursday in November rather than the last Thursday, meaning in some years one week earlier, in order to lengthen the Christmas shopping season. Most people adopted the President's change, which was later reinforced by an act of Congress, but many continued to celebrate Thanksgiving on the traditional date. Some started referring to the new date as Franksgiving.


title: Black Friday (shopping)

text: In 2015, Amazon.com was the first to offer "Black Friday in July" deals on what they called "Prime Day", promising better deals than on Black Friday. Amazon repeated the practice in 2016 and 2017, and other companies began offering similar deals.


title: Black Friday (shopping)

text: Analyst Marshal Cohen of The NPD Group claimed in 2020 that Black Friday is declining in favor of online shopping, and that the COVID-19 pandemic has accelerated this process. The pandemic also resulted in holiday deals being offered over a longer period of time, even as early as October. Fewer people shopped in person on Black Friday 2020, and most business took place online. Market research company Numerator said sellers of clothing, tools and other items considered nonessential during lockdowns were not promoted as heavily because lower production meant less available to sell. Adobe Analytics reported that online sales reached $9 billion in 2020, 22% more than the previous year. Foot traffic to stores fell 48% in 2020 from last year, according to RetailNext, while Sensormatic Solutions reported a 52% decrease.


title: Black Friday (shopping)

text: For many years, retailers pushed opening times on Black Friday earlier and earlier, eventually reaching midnight, before opening on the evening of Thanksgiving. In 2009, Kmart manager Freddy Moss opened at 7 P.M. on Thanksgiving in order to allow shoppers to avoid Black Friday traffic and return home in time for dinner with their families. Two years later, a number of retailers began opening at 8 P.M. or 9 P.M., on what became derisively known as "Black Thursday". In subsequent years, other stores have followed this trend, opening earlier and earlier on Thanksgiving, or remaining open all day, beginning in the early morning hours. Some retail and media sources have used the terms "Gray Thursday" or "Brown Thursday" instead.


title: Black Friday (shopping)

text: The 2014 "Black Thursday" sales were generally a failure, as overall sales for the holiday week-end fell 11% compared to the previous year despite heavy traffic at the stores on Thanksgiving night. In response, a number of retailers decided to go back to closing on Thanksgiving for 2015, and Walmart, although its holding firm opening on the holiday and holding its sale, also pledged to offer the same deals online for those who wished to stay home.


title: Black Friday (shopping)

text: Retailers have received pushback from some consumers over opening on Thanksgiving Day. Shopper opposition to stores opening on Thursday includes the perceived over-commercialization of Thanksgiving, retail workers not being able to spend time with their loved ones on the holiday, and Thursday doorbuster sales forcing consumers to sacrifice enjoying Thanksgiving evening with their families in order to avoid missing out on highly sought-after items which might not be available again prior to Christmas.


title: Black Friday (shopping)

text: Most retailers abandoned efforts to hold doorbuster sales on Thanksgiving in 2020; large crowds have been forbidden under most circumstances since March due to the ongoing COVID-19 pandemic, major retailers such as Walmart and Target had already reduced their hours and dropped 24/7 operations in response to the pandemic, and several retailers known for opening on the holiday (particularly Kmart, which has typically been open regular hours) have rapidly declined. According to Adobe Analytics, online shopping set a record on Thanksgiving 2020 with $5.1 billion in total spending, 21.5% higher than in 2019. Most major retailers again closed on Thanksgiving in 2021, with Target stating that its decision would be permanent; the small minority of retail chains remaining open on the holiday that year were limited mainly to pharmacies, dollar stores and grocery stores, retail categories that traditionally do not hold doorbusters.


title: Black Friday (shopping)

text: Black Friday is not an official holiday in the United States, but California and some other states observe "The Day After Thanksgiving" as a holiday for state government employees. It is sometimes observed in lieu of another federal holiday, such as Columbus Day. Many non-retail employees and schools have both Thanksgiving and the following Friday off. Along with the following regular week-end, this makes Black Friday weekend a four-day weekend, which is said to increase the number of potential shoppers.


title: Black Friday (shopping)

text: The SouthPark neighborhood of Charlotte, North Carolina, is the most trafficked area of the United States on Black Friday.


title: Black Friday (shopping)

text: Black Friday is a shopping day for a combination of reasons. As the first day after the last major holiday before Christmas, it marks the unofficial beginning of the Christmas shopping season. Additionally, many employers give their employees the day off as part of the Thanksgiving holiday week-end. In order to take advantage of this, virtually all retailers in the country, big and small, offer various sales including limited amounts of "doorbuster" items to entice traffic.


title: Black Friday (shopping)

text: For many years, it was common for retailers to open at 6a.m., but in the late 2000s many opened at 4a.m.–5a.m. The early 2010s have seen retailers extend beyond normal hours in order to maintain an edge or to simply keep up with the competition. In 2010, Toys 'R' Us began their Black Friday sales at 10p.m. on Thanksgiving and further upped the ante by offering free boxes of Crayola crayons and coloring books for as long as supplies lasted. Other retailers, like Sears, Express, MK, Victoria's Secret, Zumiez, Tillys, American Eagle Outfitters, Nike, Jordan, Puma, Aéropostale, and Kmart, began Black Friday sales early Thanksgiving morning and ran them through as late as 11p.m. Friday evening. Forever 21 went in the opposite direction, opening at normal hours on Friday, and running late sales until 2a.m. Saturday morning. In 2011, when several retailers (including Target, Kohl's, Macy's, Best Buy, and Bealls) opened at midnight for the first time. In 2012, Walmart and several other retailers announced that they would open most of their stores at 8p.m. on Thanksgiving, prompting calls for a walkout among some workers. In 2014, stores such as JCPenney, Best Buy, and Radio Shack opened at 6a.m. on Thanksgiving while stores such as Target, Walmart, Belk, and Sears opened at 7p.m. on Thanksgiving. Three statesRhode Island, Maine, and Massachusettsprohibit large supermarkets, big box stores, and department stores from opening on Thanksgiving, in what has been referred to as a legacy of blue laws. A bill to allow stores to open on Thanksgiving was the subject of a public hearing on July 8, 2017.


title: Black Friday (shopping)

text: Historically, it was common for Black Friday sales to extend throughout the following week-end. However, this practice has largely disappeared in recent years, perhaps because of an effort by retailers to create a greater sense of urgency.


title: Black Friday (shopping)

text: The news media usually give heavy play to reports of Black Friday shopping and their implications for the commercial success of the Christmas shopping season, but the relationship between Black Friday sales and retail sales for the full holiday season is quite weak and may even be negative.


title: Black Friday (shopping)

text: In 2014, spending volume on Black Friday fell for the first time since the 2008 recession. 50.9 billion dollars was spent during the four-day Black Friday week-end, down 11% from the previous year. However, the U.S. economy was not in a recession. Christmas creep has been cited as a factor in the diminishing importance of Black Friday, as many retailers now spread out their promotions over the entire months of November and December rather than concentrate them on a single shopping day or week-end.


title: Black Friday (shopping)

text: On April 23, 2014, "".blackfriday"" joined a growing list of ICANN top-level domains (such astraditionally.com, .net, and .org).


title: Black Friday (shopping)

text: In 2015, Neil Stern of McMillan Doolittle said, "Black Friday is quickly losing its meaning on many fronts," because many stores opened on Thanksgiving, and a lot of sales started even earlier than that. Online shopping also made the day less important. A Gallup poll in 2012 has shown that only 18% of American adults planned to shop during Black Friday.


title: Black Friday (shopping)

text: In 2020, an article in "Ad Age" magazine stated that "an American capitalist tradition ... has been on the wane for years as online shopping rises in popularity" but the COVID-19 pandemic, "which has dramatically altered shopping patterns, has seemingly dealt a fatal blow."


title: Black Friday (shopping)

text: By 2021, spending during the holiday season was expected to continue to increase, but Black Friday was no longer a single day. It was instead an opportunity for retailers to offer deals during the season as online shopping continued to change consumer behavior. COVID-19 had increased the changes in buyer and retailer moves toward online transactions. Supply chain disruptions caused consumers to buy earlier when they could find items.


title: Black Friday (shopping)

text: The large population centers on Lake Ontario and the Lower Mainland in Canada have always attracted cross-border shopping into the United States, and as Black Friday became more popular in the U.S., Canadians often flocked over the border because of their lower prices and a stronger Canadian dollar. After 2001, many were traveling for the deals across the border. Starting in 2008 and 2009, due to the parity of the Canadian dollar compared with the American dollar, several major Canadian retailers ran Black Friday deals of their own to discourage shoppers from leaving Canada.


title: Black Friday (shopping)

text: The year 2012 saw the biggest Black Friday to date in Canada, as Canadian retailers embraced it in an attempt to keep shoppers from travelling across the border.


title: Black Friday (shopping)

text: Before the advent of Black Friday in Canada, the most comparable holiday was Boxing Day in terms of retailer impact and consumerism. Black Fridays in the U.S. seem to provide deeper or more extreme price cuts than Canadian retailers, even for the same international retailer.


title: Black Friday (shopping)

text: In the United Kingdom, the term "Black Friday" originated within the Police and NHS to refer to the Friday before Christmas. It is the day when emergency services activate contingency plans to cope with the increase in workload due to many people going out drinking on the last Friday before Christmas. Contingencies can include setting up mobile field hospitals near City Centre nightspots. The term has then been adopted outside those services to refer to the evening and night of the Friday immediately before Christmas, and would now be considered a mainstream term and not simply as jargon of the emergency services.


title: Black Friday (shopping)

text: Traditionally, Boxing Day had been considered the biggest shopping day of the year in the UK. In the 2010s, several American-owned retailers such as Amazon UK and the Walmart-owned chain Asda, began to hold U.S.-style Black Friday promotions; in 2014, more British retailers began to adopt the concept, including Argos, John Lewis, and Very. That year, police forces were called to shops across Britain to deal with crowd control issues, assaults, threatening customers, and traffic issues. In response to incidents at branches of Tesco, Greater Manchester Police's deputy chief constable Ian Hopkins said shoppers had behaved in an "appalling" fashion, and criticized shops for not making adequate security arrangements to ensure the safety of customers." Following these incidents, some retailers began to discontinue or heavily modify their promotions, with Asda stating that it would not hold all of its sales across a single day.


title: Black Friday (shopping)

text: In 2016, total spending on online retail sites on Black Friday was £1.23 billion, a 2.2% year-over-year increase over 2015. In 2017, UK retail sales in November grew faster than in December for the first time.


title: Black Friday (shopping)

text: In Mexico, Black Friday was the inspiration for the government and retailing industry to create an annual week-end of discounts and extended credit terms, El Buen Fin, meaning "the good weekend" in Spanish. El Buen Fin has been in existence since 2011 and takes place on November in the week-end prior to the Monday in which the Mexican Revolution holiday is pushed from its original date of November 20, as a result of the measure taken by the government of pushing certain holidays to the Monday of their week in order to avoid the workers and students to make a "larger" week-end (for example, not attending in a Friday after a Thursday holiday, thus making a four-day week-end). On this week-end, major retailers extend their store hours and offer special promotions, including extended credit terms and price promotions.


title: Black Friday (shopping)

text: The concept was imported in Romania by and Flanco in 2011 and became bigger each year. The two reported the biggest Black Friday sales in 2014. eMAG sold products worth some 37 million euros while Flanco's sales totaled 22 million euros. Hundreds of retailers announced their participation in the 2015 campaign.


title: Black Friday (shopping)

text: In 2015, 11 million Romanians say they have heard about Black Friday which is 73% of the 15 million people target segment. 6.7 million plan on buying something on biggest shopping event of the year in Romania.


title: Black Friday (shopping)

text: The holiday shopping season in India has traditionally been aligned around the "festive period" of major festivals usually falling around October, such as Diwali. Seeking to emulate U.S.-style shopping events such as Black Friday, online retailers adopted the practice of holding multi-day promotions during this period, such as Amazon's "Great Indian Festival" and Flipkart's "Big Billion Days". India's Independence Day (15 August) had also recently attracted similar events.


title: Black Friday (shopping)

text: Nonetheless, the concept of Black Friday has also been imported into the subcontinent via international retailers–a move that influenced some Indian retailers to also adopt the promotion.


title: Black Friday (shopping)

text: French businesses are slowly introducing the Black Friday custom into the market. Discounts of up to 85% were given by retailing giants such as Apple and Amazon in 2014. French electronics retailers such as FNAC and Auchan advertised deals online, while Darty also took part in this once-a-year monster sale. Retailers favored the very American term "Black Friday" to "Vendredi noir" in their advertisements. In 2016, because of the terror attacks in Paris in November the year before, some retailers used the name "Jour XXL" (XXL day) instead of Black Friday. An alternative was brought up by some online businesses in 2018, called "French Days", which goal is to replicate Black Friday during spring season (starting around the first day of May).


title: Black Friday (shopping)

text: On November 20, 2020, the French government finalized an agreement with e-commerce businesses like Amazon and supermarket chains to postpone Black Friday promotions by a week. Discounted shopping promotions were to begin on December 4 instead, after physical stores shuttered during the COVID-19 pandemic were allowed to reopen.


title: Black Friday (shopping)

text: In Germany, "Black Friday" retailer advertisements refer to "Black Week" and "Black Shopping" in English with sales lasting an entire week (excluding Sundays when most retail stores are closed). During this sales period, stores keep their normal working hours. Although goods are offered at reduced prices, the prices are not cut significantly more than normal weekly price reductions. Apple was the first company to run a special Black Friday campaign for the German market in 2006. Apple never used the name Black Friday in Germany, but promotes only a "one-day shopping event". In the first years, mostly internet retailers have used the event as an occasion to attract new customers with discounts, but bricks and mortar stores have already begun to adapt the shopping event. For the first time ever, German customers spent more than €1 billion during the Black Friday week-end in 2016: According to a Centre for Retail Research study, German customers spend around €1.3 billion ($1.54 billion) during the four days from Black Friday to Cyber Monday 2016.


title: Black Friday (shopping)

text: In Germany the term "Black Friday" was registered as a wordmark in December 2013. The owner is Super Union Holdings Limited, a Chinese company. The trademark rights were largely revoked by the German Federal Patent Court and the German Federal Court of Justice. The trademark protection has thus essentially lapsed.


title: Black Friday (shopping)

text: In 2015, Swiss retailer Manor was the first to launch a special Black Friday promotion. The year after, most Swiss retailers launched special offers during the Black Friday Week. It is estimated that customers spent around 400 million Swiss Francs on Black Friday 2018. In recent years, Singles Day got more and more important in Switzerland. This shopping day could replace Black Friday as the most important shopping day in Switzerland in 2019


title: Black Friday (shopping)

text: In Australia the term is controversial, in Australia Black Friday refers not to shopping at all but to the devastating Black Friday bushfires which occurred in Victoria 1938–39. It wasn't until the 2010s that it was heavily promoted as a shopping day in Australia by in-store and online retailers, despite backlash and confusion by consumers. In 2011, Online Shopping USA hosted an event on Twitter. Twitter users had to use the hashtag #osublackfriday, which allowed them to follow along and tweet their favourite deals and discounts from stores. In 2013, Apple extended its Black Friday deals to Australia. Purchasing online gave customers free shipping and free iTunes gift cards with every purchase. The deals were promoted on its website, reading "Official Apple StoreOne day Apple shopping event Friday, November 29". Australia Post's ShopMate parcel-forwarding service allows Australian customers to purchase products with "Black Friday" deals from the US and get them shipped to Australia. In addition to this, numerous stores in the country run Black Friday promotions in-store and online throughout the country.


title: Black Friday (shopping)

text: Black Friday started picking up in New Zealand around 2013. In 2015, major retailers such as The Warehouse, Noel Leeming and Harvey Norman offered Black Friday sales, and by 2018 were joined by Farmers, JB Hi-Fi, Briscoes and Rebel Sport. Paymark, which processes around 75% of New Zealand's electronic transactions, recorded $219 million NZD (US$151 million) of transactions on Black Friday 2017, up over 10% from the previous year.


title: Black Friday (shopping)

text: In Norway, Black Friday started as a publicity stunt campaign back in 2010 to increase the sales to the shopping mall Norwegian Outlet. Since the introduction, it has been promoted every year in a larger and growing market all over the country.


title: Black Friday (shopping)

text: Black Friday is known as "Viernes Negro" in Costa Rica. In Panama, Black Friday was first celebrated in 2012, as a move from the Government to attract local tourism to the country's capital city. During its first year, it was believed to have attracted an inflow of about 35,000 regional tourists according to the government's immigration census.


title: Black Friday (shopping)

text: In Liechtenstein and Switzerland, Black Friday Sale is a joint sales initiative by hundreds of online vendors. Over its first 24-hour run on November 28, 2013, more than 1.2 million people visited the site, making it the single largest online shopping event in German-speaking countries. There has been growing interest for Black Friday in Poland as well.


title: Black Friday (shopping)

text: 2014 marked the introduction in Bolivia, Colombia, Denmark, Italy, Finland, France, Ireland, Lebanon, Nigeria, South Africa and Sweden.


title: Black Friday (shopping)

text: For the Middle East, U.A.E. Black Friday started as White Friday campaign in 2014. In 2018 local e-commerce platform noon.com created Yellow Friday in the U.A.E. and Saudi Arabia. The Yellow Friday Sale is now an annual event in Saudi Arabia, Iran and the U.A.E., falling around the same time as Black Friday globally.


title: Black Friday (shopping)

text: In 2015, Spain joined with some small retailers. The celebration became more famous year by year, until the big retailers grew.


title: Black Friday (shopping)

text: In the Netherlands, Black Friday was seriously introduced in 2015. Some years before, there were already a number of large and small retailers that used Black Friday in their marketing. However, with a total of 35 participating stores, 2015 can be considered the year in which Black Friday started in the Netherlands due to a more widespread support of large retailers. The popularity of Black Friday has grown rapidly in the Netherlands. The number of participating stores has increased to over 125 during the Black Friday period of 2017. For the 2018 edition, 166 shops joined the largest black Friday platform in the Netherlands.


title: Black Friday (shopping)

text: Black Friday in Belgium is seriously marketed by retailers since 2016. Especially online shops have broke sales records during the last edition of Black Friday, which provides a base for further growth of popularity of Black Friday in Belgium. After 2016, Black Friday in Belgium has grown strongly. The participating shops have increased to over seventy during the Black Friday period of 2017. During Black Friday 2018, a total of 119 participating stores were measured in Belgium.


title: Black Friday (shopping)

text: In 2017, Black Friday became widely popular in Latvia. There was even a Black week and Black week-end sales in shopping centres.


title: Black Friday (shopping)

text: Black Friday has been increasingly adopted by stores in Brazil since 2010, although not without its share of inflated prices and other scams, especially in its earlier years, earning the nickname ""Black Fraude"" (Black Fraud) or also ""Black Furadei"", which comes from the slang word ""furada"", meaning a "jam" or tough situation, usually involving money. It is also common to hear Brazilian people say that prices on Brazilian Black Friday are "half of the double". However, currently, the term "Black Friday" has become so popular in the country that stores have been under closer scrutiny from consumers and cases of known scams have been reduced greatly.


title: Black Friday (shopping)

text: Despite frequent attempts to control the crowds of shoppers, minor injuries are common among the crowds, usually as a result of being pushed or thrown to the ground in small stampedes. While most injuries remain minor, serious injuries and even deliberate violence have taken place on some Black Fridays.


title: Black Friday (shopping)

text: In 2008, a crowd of approximately 2,000 shoppers in Valley Stream, New York, waited outside for the 05:00 opening of the local Walmart. As opening time approached, the crowd grew anxious and when the doors were opened, the crowd pushed forward, breaking the door down, and 34-year-old employee Jdimytai Damour was trampled to death. The shoppers did not appear concerned with the victim's fate, expressing refusal to halt their stampede when other employees attempted to intervene and help the injured employee, complaining that they had been waiting in the cold and were not willing to wait any longer. Shoppers had begun assembling as early as 21:00 the evening before. Even when police arrived and attempted to render aid to the injured man, shoppers continued to pour in, shoving and pushing the officers as they made their way into the store. Several other people incurred minor injuries, including a pregnant woman who had to be taken to the hospital. The incident may be the first case of a death occurring during Black Friday sales; according to the National Retail Federation, "We are not aware of any other circumstances where a retail employee has died working on the day after Thanksgiving."


title: Black Friday (shopping)

text: On the same day, two people were fatally shot during an altercation at a Toys 'R' Us in Palm Desert, California.


title: Black Friday (shopping)

text: During Black Friday 2010, a Madison, Wisconsin woman was arrested outside of a Toys 'R' Us store after cutting in line, and threatening to shoot other shoppers who tried to object. A Toys for Tots volunteer in Georgia was stabbed by a shoplifter. An Indianapolis woman was arrested after causing a disturbance by arguing with other Wal-Mart shoppers. She had been asked to leave the store, but refused.


title: Black Friday (shopping)

text: A man was arrested at a Florida Wal-Mart on drug and weapons charges after other shoppers waiting in line for the store to open noticed he was carrying a handgun and reported it to the police. He was discovered to also be carrying two knives and a pepper spray grenade. A man in Buffalo, New York, was trampled when doors opened at a Target store and unruly shoppers rushed in, in an episode reminiscent of the deadly 2008 Wal-Mart stampede.


title: Black Friday (shopping)

text: On Black Friday 2011, a woman at a Porter Ranch, California Walmart used pepper spray on fellow shoppers, causing minor injuries to a reported 20 people who had been waiting hours for the store to open. The incident started as people waited in line for the newly discounted Xbox 360. A witness said a woman with two children in tow became upset with the way people were pushing in line. The witness said she pulled out pepper spray and sprayed the other people in line. Another account stated: "The store had brought out a crate of discounted Xbox 360s, and a crowd had formed to wait for the unwrapping, when the woman began spraying people 'in order to get an advantage,' according to the police. In an incident outside a Walmart store in San Leandro, California, one man was wounded after being shot following Black Friday shopping at about 1:45 a.m. A 61-year-old pharmacist collapsed and was left for dead by shoppers while being trampled and passed by a stampede. He died soon after from his injuries.


title: Black Friday (shopping)

text: On Black Friday 2012, two people were shot outside a Wal-Mart in Tallahassee, Florida, during a dispute over a parking space.


title: Black Friday (shopping)

text: On Black Friday in 2013, a person in Las Vegas who was carrying a big-screen TV home from a Target store on Thanksgiving was shot in the leg as he tried to wrestle the item back from a robber who had just stolen it from him at gunpoint. In Romeoville, Illinois, a police officer shot a suspected shoplifter driving a car that was dragging a fellow officer at a Kohl's department store. The suspect and the dragged officer were treated for shoulder injuries. Three people were arrested. In another situation, a 29-year-old shopper was arrested in a Walmart in New Jersey after arguing with a store manager about a TV and attacking an officer. He was charged with disorderly conduct, aggravated assault, and resisting arrest


title: Black Friday (shopping)

text: In 2014, three buyers were arrested after a group of five people started fighting at a Kohl's store in Tustin, California. Two female victims were found with facial lacerations, and one of them was taken to hospital with minor injuries, while the other was released on scene. According to officials, three other females were suspects for the assault and were taken into custody. Two people were arrested after a brawl on Black Friday at a northwest side mall in Indianapolis. In Los Angeles, two women were fighting at a Walmart in Norwalk, California, over a Barbie doll on Thanksgiving night.


title: Black Friday (shopping)

text: Several people fighting at a mall in Florence, Kentucky, allegedly over a pair of Air Jordan sneakers. This year was called "The worst Black Friday brawls in history" at that time due to the heavy use of smartphones that could instantly capture video.


title: Black Friday (shopping)

text: In 2016, 21-year-old Demond Cottman was shot and killed around 01:00 Friday morning outside a Macy's store in New Jersey. The shooter fired multiple shots, leaving an SUV covered in bullet holes, but the motives remain unclear. Cottman's 26-year-old brother was also injured. A shooting at the Wolfchase Galleria Mall in Memphis, Tennessee, left one man injured. Derrick Blackburn, 19, was later arrested for unlawful possession of a weapon.


title: Black Friday (shopping)

text: In Vancouver, a shirtless man attacked several people using his belt as a whip. The incident occurred outside an Adidas store where a crowd was awaiting the release of a rare shoe.


title: Black Friday (shopping)

text: At the Riverchase Galleria in Hoover, Alabama, Emantic Fitzgerald Bradford Jr., was shot and killed by a security guard after two people were wounded in a shooting. On Saturday, the police announced that the shooter was not Bradford, but he was involved in the shooting.


title: Black Friday (shopping)

text: A fight led to a shooting in the food court of the Destiny USA mall in Syracuse, New York. The mall went into lockdown until shoppers and staff were released starting at about 8:00 P.M. with all shopping activity suspended. 21-year-old Kyree Truax was arrested and charged with second-degree assault, second-degree criminal possession of a weapon, and second-degree reckless endangerment for shooting the victim twice in the leg.


title: Black Friday (shopping)

text: Some online stores invest a lot of money in promotional campaigns to generate more sales and drive traffic to their stores. However, they often forget about the high loads their sites are going to experience. According to "Retail Gazette", "A number of major retailers' websites went down as they failed to cope with the surge in Black Friday traffic in 2017... This just highlights that some retailers have not taken the necessary steps to prepare for Black Friday. Failing to prepare for peak can cause poor performance, site downtime, and ultimately lost revenue for retailers". Such carelessness results in huge reputational damage. Moreover, The 2017 Veeam Availability Report shows that "Unplanned downtime costs organisations around the world an average of R270m annually, up from the R210m of the previous year".


title: Black Friday (shopping)

text: Some websites offer information about day-after-Thanksgiving specials up to a month in advance. The text listings of items and prices are usually accompanied by pictures of the actual ad circulars. These are either leaked by insiders or intentionally released by large retailers to give consumers insight and allow them time to plan.


title: Black Friday (shopping)

text: In recent years, some retailers (including Walmart, Target, OfficeMax, Big Lots, and Staples) have claimed that the advertisements they send in advance of Black Friday and the prices included in those advertisements are copyrighted and are trade secrets. Some of these retailers have used the take-down system of the Digital Millennium Copyright Act as a means to remove the offending price listings.


title: Black Friday (shopping)

text: The benefit of threatening Internet sites with a DMCA based lawsuit has proved tenuous at best. While some sites have complied with the requests, others have either ignored the threats or simply continued to post the information under the name of a similar-sounding fictional retailer. However, careful timing may mitigate the take-down notice. An Internet service provider in 2003 brought suit against Best Buy, Kohl's, and Target Corporation, arguing that the take-down notice provisions of the DMCA are unconstitutional. The court dismissed the case, ruling that only the third-party posters of the advertisements, and not the ISP itself, would have standing to sue the retailers.


title: Black Friday (shopping)

text: Usage of Black Friday Advertising Tip sites and buying direct varies by state in the U.S., influenced in large part by differences in shipping costs and whether a state has a sales tax. However, in recent years, the convenience of online shopping has increased the number of cross-border shoppers seeking bargains from outside of the U.S., especially from Canada. Statistics Canada indicates that online cross-border shopping by Canadians has increased by about 300M a year since 2002. The complex nature of additional fees such as taxes, duties, and brokerage can make calculating the final cost of cross-border Black Friday deals difficult. Cross-border shopping solutions exist to mitigate the problem through estimation of the various cost involved.


title: Black Friday (shopping)

text: In 2019, Adobe shopping data showed that around 39% of the shopping was done through smartphones. In 2022, 48% of online sales are made through smartphones, up from 44% in 2021. Meanwhile, consumers spent a record $9.12 billion shopping online during Black Friday this year.


title: Black Friday (shopping)

text: The term "Cyber Monday", a neologism invented in 2005 by the National Retail Federation's division Shop.org, refers to the Monday immediately following Black Friday based on a trend that retailers began to recognize in 2003 and 2004. Retailers noticed that many consumers, who were too busy to shop over the Thanksgiving week-end or did not find what they were looking for, shopped for bargains online that Monday from home or work. In 2010, Hitwise reported:


title: Black Friday (shopping)

text: In 2013, Cyber Monday online sales grew by 18% over the previous year, hitting a record $1.73 billion, with an average order value of $128.


title: Black Friday (shopping)

text: In 2014, Cyber Monday was the busiest day of the year with sales exceeding $2 billion in desktop online spending, up 17% from the previous year.


title: Black Friday (shopping)

text: As reported in the "Forbes" "Entrepreneurs" column on December 3, 2013: "Cyber Monday, the online counterpart to Black Friday, has been gaining unprecedented popularityto the point where Cyber Sales are continuing on throughout the week." Peter Greenberg, Travel Editor for CBS News, further advises: "If you want a real deal on Black Friday, stay away from the mall. Black Friday and Cyber Monday are all part of Cyber Week..."


title: Black Friday (shopping)

text: The National Retail Federation releases figures on the sales for each Thanksgiving week-end. The Federation's definition of "Black Friday week-end" includes Thursday, Friday, Saturday and projected spending for Sunday. The survey estimates number of shoppers, not number of people.


title: Black Friday (shopping)

text: The length of the shopping season is not the same across all years: the date for Black Friday varies between November 23 and 29, while Christmas Eve is fixed at December 24.


title: Black Friday (shopping)

text: These are various day-long events similar to Black Friday around the world or any other events on the same day as Black Friday.


title: Gacha game

text: A is a video game that implements the (toy vending machine) mechanic. Similar to loot boxes, gacha games entice players to spend in-game currency to receive a random in-game item. Some in-game currency generally can be gained through game play, and some by purchasing it from the game publisher using real-world funds.


title: Gacha game

text: The game model began to be widely used in the early 2010s, particularly in Japan. Most of the highest-grossing mobile games in Japan use it, and it has become an integral part of Japanese mobile game culture. The game mechanism is also increasingly used in Chinese and Korean games, as well as Western games. Despite their ubiquity, "gacha" games have been criticized for being addictive, and are often compared to gambling due to the incentive to spend real-world money on chance-based rewards.


title: Gacha game

text: There are many collectable characters, cards, or other items (details will vary based on nature of game). Many of them are obtainable only through a "gacha" mechanic. In this, the player "pulls" or "spins" in a manner analogous to a slot machine or roulette wheel. In doing so they expend a fixed portion of a premium currency in exchange for receiving a random "drop" from the banner "rolled on". Some of the rewards drop less frequently than others. It is common for the schema of item rarities to be public information, dubbed "open gacha". It is common for there to be a rarity tier on around the order of appearing in one percent of rolls. Between this rarity and the commonality of limited-time availability of promoted gacha drops, players are encouraged to roll the while their desired item is available.


title: Gacha game

text: Some gacha models use a pity system: the player will be guaranteed an item after pulling for that item a large number of times without success. "Soft" pity increases the probability slightly of getting a rare item with every pull, counting up and recalculating the probability until the rare item is received, while "hard" pity uses a counter to keep track of the number of pulls and automatically dispense the rare item after reaching a preset number of rolls.


title: Gacha game

text: Games can include multiple in-game currencies with complex schemes for converting between them. This makes it more difficult for the player to model the dollar cost of a unit of a currency that isn't the "premium" one.


title: Gacha game

text: It is common for it to be possible to get even the "premium" (obtained with real money) currency through gameplay, albeit in rigorously limited quantities.


title: Gacha game

text: Many kinds of virtual items can be in the loot table for a banner. Gameplay units such as cards, characters, equippable gear, or more abstract loot such as "experience" are all possible.


title: Gacha game

text: In many games, rewards are essential for players to make progress in the game. Players are generally given free or discounted in low amounts on a regular schedule, in exchange for logging in or doing in-game tasks.


title: Gacha game

text: Banners are "pools" of available items (characters, loot, cards, etc) that players can "roll" on. Offered banners can be perpetually available or can have a limited duration. Games generally have some of both, with player retention efforts and in-game advertising emphasizing the limited availability of some or all of the items in the latter.


title: Gacha game

text: Stamina is a resource that is required for, and consumed by, core in-game actions such as (in a fighting-oriented game) beginning combat encounters. It regenerates over time, often up to a cap. It can typically be regenerated or gained instantly through some form of microtransaction or premium currency spending.


title: Gacha game

text: Game developers have praised as a free-to-play monetization strategy. Most developers that work primarily with free-to-play games recommend it be incorporated into the game starting with the concept for maximum monetization potential.