Abc conjecture
The "abc" conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Oesterlé (in 1988) and Masser (in 1985). It is stated in terms of three positive integers, "a", "b" and "c" (hence the name) that are relatively prime and satisfy "a" + "b" = "c". If "d" denotes the product of the distinct prime factors of "abc", the conjecture essentially states that "d" is usually not much smaller than "c". In other words: if "a" and "b" are composed from large powers of primes, then "c" is usually not divisible  by large powers of primes. A number of famous conjectures and theorems in number theory would follow immediately from the "abc" conjecture or its versions. American mathematician Dorian Goldfeld described the "abc" conjecture as "The most important unsolved problem in Diophantine analysis". The "abc" conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the "abc" conjecture. The "abc" conjecture was shown to be equivalent to the modified Szpiro's conjecture. Various attempts to prove the "abc" conjecture have been made, but  none are currently accepted by the mainstream mathematical community and as of 2020, the conjecture is still regarded as unproven. Formulations. Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integer "n", the radical of "n", denoted rad("n"), is the product of the distinct prime factors of "n". For example If "a", "b", and "c" are coprime positive integers such that "a" + "b" = "c", it turns out that "usually" "c" < rad("abc"). The "abc conjecture" deals with the exceptions. Specifically, it states that: An equivalent formulation is: A third  equivalent formulation of the conjecture involves the "quality" "q"("a", "b", "c") of the triple ("a", "b", "c"), which is defined as For example: A typical triple ("a", "b", "c") of coprime positive integers with "a" + "b" = "c" will have "c" < rad("abc"), i.e. "q"("a", "b", "c") < 1. Triples with "q" > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. The third formulation is: Whereas it is known that there are infinitely many triples ("a", "b", "c") of coprime positive integers with "a" +  b = "c" such that "q"("a", "b", "c") > 1, the conjecture predicts that only finitely many of those have "q" > 1.01 or "q" > 1.001 or even "q" > 1.0001, etc. In particular, if the conjecture is true, then there must exist a triple ("a", "b", "c") that achieves the maximal possible quality "q"("a", "b", "c") . Examples of triples with small radical. The condition that "ε" > 0 is necessary as there exist infinitely many triples "a", "b", "c" with "c" > rad("abc"). For example, let The integer "b" is divisible by 9: Using this fact, the following  calculation is made: By replacing the exponent 6"n" by other exponents forcing "b" to have larger square factors, the ratio between the radical and "c" can be made arbitrarily small. Specifically, let "p" > 2 be a prime and consider Now it may be plausibly claimed that "b" is divisible by "p": The last step uses the fact that "p" divides 2 − 1. This follows from Fermat's little theorem, which shows that, for "p" > 2, 2 = "pk" + 1 for some integer "k". Raising both sides to the power of "p" then shows that 2 =  "p"(...) + 1. And now with a similar calculation as above, the following results: A list of the highest-quality triples (triples with a particularly small radical relative to "c") is given below; the highest quality, 1.6299, was found by Eric Reyssat for Some consequences. The "abc" conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a conditional proof. The consequences include: Theoretical results. The "abc" conjecture implies that "c" can be bounded above by a near -linear function of the radical of "abc". Bounds are known that are exponential. Specifically, the following bounds have been proven: In these bounds, "K" and "K" are constants that do not depend on "a", "b", or "c", and "K" is a constant that depends on "ε" (in an effectively computable way) but not on "a", "b", or "c". The bounds apply to any triple for which "c" > 2. Computational results. In 2006, the Mathematics Department of Leiden University in the Netherlands, together with the Dutch Kennislink science institute, launched the ABC@Home project, a grid computing system, which aims to  discover additional triples "a", "b", "c" with rad("abc") < "c". Although no finite set of examples or counterexamples can resolve the "abc" conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally. As of May 2014, ABC@Home had found 23.8 million triples. Note: the "quality" "q"("a", "b", "c") of the triple ("a", "b", "c") is defined above. Refined forms, generalizations and related statements. The "abc" conjecture is an integer analogue of the Mason–Stothers theorem for polynomials. A strengthening, proposed by , states that in  the "abc" conjecture one can replace rad("abc") by where "ω" is the total number of distinct primes dividing "a", "b" and "c". Andrew Granville noticed that the minimum of the function formula_3 over formula_4 occurs when formula_5 This inspired to propose a sharper form of the "abc" conjecture, namely: with "κ" an absolute constant. After some computational experiments he found that a value of formula_6 was admissible for "κ". This version is called the "explicit "abc" conjecture". where Ω("n") is the total number of prime factors of "n", and where Θ("n") is the number of integers up to "n" divisible  only by primes dividing "n". Let "k" = rad("abc"). They conjectured there is a constant "C" such that holds whereas there is a constant "C" such that holds infinitely often. Claimed proofs. Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards. Since August 2012, Shinichi Mochizuki has claimed a proof of Szpiro's conjecture and therefore the "abc" conjecture. He released a series of four preprints developing a new theory he called inter-universal Teichmüller theory (IUTT), which is then applied to prove the "abc" conjecture. The papers have not been accepted by  the mathematical community as providing a proof of "abc". This is not only because of their length and the difficulty of understanding them, but also because at least one specific point in the argument has been identified as a gap by some other experts. Although a few mathematicians have vouched for the correctness of the proof, and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large. In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki. While they did not resolve the differences, they  brought them into clearer focus. Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that ... small modifications will not rescue the proof strategy"; Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications. On April 3, 2020, two mathematicians from the Kyoto research institute where Mochizuki works announced that his claimed proof would be published in "Publications of the Research Institute for Mathematical Sciences", the institute's journal. Mochizuki is chief editor of the journal but recused himself from  the review of the paper. The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by "Nature" as "unlikely to move many researchers over to Mochizuki's camp". In March 2021, Mochizuki's proof was published in RIMS.