Prime numbers

A is an integer greater than 1 that has no divisors other than 1 and itself.
== List of notable prime numbers ==
*2 is the smallest prime and the only even prime.
*3 is the only number that is both a Mersenne prime and a Fermat prime. It is also the first odd prime.
*5 is the second Fermat prime.
*7 is the second Mersenne prime. It's also the smallest cuban prime.
*17 is the third Fermat prime and the third .
*31 is the third Mersenne prime.
*101 is the smallest 3-digit prime. It's also a twin prime with 103. 101 is also a , a , and the largest known prime in the form 10n+1.Prime Curios!: 101
*113 is a .OEIS, Sequence A005384 (accessed 2020-11-22)
**It is also is the sum of the hyperfactorials of the first three positive numbers.
**Furthermore, \(\frac{355}{113}\) is a famous approximation of \(\pi\) named . It's equal to 3.141592920...
*131 is a palindromic .
*137 is a and the fifth largest known Stern prime.
**It is also approximately the reciprocal of the .
**Since and are also used as AM radio carriers, there are 137 AM radio frequencies in Europe.
**With a leading zero, it is also the German mass traffic telephone number prefix.
*151 is a centered decagonal number and a palindromic prime.
**It is also the last number n, such that en is smaller than the first noncanonical -illion.
**Furthermore, in Orthodox churches, the Book of contains 151 psalms.
**Finally, it is also the number of species in the first generation.
*163 is the largest . For this reason, it has been used in the Ramanujan constant.
**The McKay-Thompson series of span a 163- .
**It is also the number of white in a set.
*The 167 is the first number n, such that 4n is larger than a googol.
**It is also the number of hours in the spring transition .
**And in , the starts at 167 MHz.
*181 is a , a , a palindromic prime and a .
*191 is a centered polygonal number, a palindromic prime and a .
**It is also the number of non-control characters.
*193 is a , a , a , and the number of ways to add seven ordinals.
*199 is a , a and a permutable prime.
**Since it is one less than 200, it is , or as a part of a price.
*211 is a centered decagonal number, a centered polygonal number and a prime .
*227 is the fourth largest known Stern prime.
**It is also the number of pips in a set.
*239 is a . It was the PEGG value on May 22nd, 2017.
**Since 2392 + 1 = 2 × 134, it also appears in many e.
**It is the largest integer which cannot be written as a ; the only other nonnegative integer with this property is 23.
**It is also one of only seven nonnegative integers which cannot be written as a sum of eighteen fourth powers; the largest integer with this property is 559.
*The number 257 is a \(2^{2^3}+1\).
**The isotope -257 is the heaviest that can be formed by from naturally occuring .
*The number 271 is a cuban prime, and the only prime house number.
*277 is a centered polygonal number and a .
*311 is a permutable prime and a right-.
*313 is a centered square number and a palindromic prime.
*317 is an index of a and a .
*337 is a left-truncatable prime, a permutable prime and a .
*353 is a palindromic Proth prime.
*373 is a palindromic permutable prime.
*383 is a palindromic prime, a Thabit prime and a .
*409 is a .
**Since it is the integral part of the quotient of the sector size (2 ) by the size, it is also the number of keys in the disk-key-block.
*The number 443 has been used in the definition of the triangrolplex.
*449 is a minimal prime, a Proth prime, and the number of ways to add eight ordinals.
*499 is a minimal prime.
**Since it is one less than 500, it is often used as a price, or as a part of a price.
*541 is the 100th prime.
*The number 563 is the largest known .
*613 is a and a left-truncatable prime.
**It is also the number of in .
*619 is an alternating factorial prime and a .
*709 is an emirp.
**A method for generating a sequence of primes is to start with 1, then choosing the smallest prime successor of a multiple of the previous number in each step. The compositeness can be easily certified by or , and the primality by . The resulting sequence starts with 1, 2, 3, 7, 29, 59, 709, … ().
**Another method for generating a sequence of primes is to start with 1, then choosing the n-th prime, where n is the previous number. But this sequence is harder to calculate. It starts with 1, 2, 3, 5, 11, 31, 127, 709, … ().
**Furthermore, it is the largest number n, for which en can be represented in the .
**Finally, it is the number of seats in the 19th , which is the largest democratically elected national parliament house ever.
*719 is a prime number. As 119, 121 and 721 are all composite, it is the only 3-digit .
**It is also the number of hours in a 30-day month (April, June, September or November) containing a spring transition.
*727 is a palindromic prime. It is also a reference to Shigetora's 727pp play choke on XI - Blue Zenith in osu! Definition Of 727 on osu!
*733 is a permutable prime and a right-truncatable prime.
*757 is a Hogben number and a palindromic prime.
*773 is a and the only 3-digit restricted left-truncatable prime.
*787 is a palindromic prime.
*The number 797 is the largest palindromic right-truncatable prime and the largest palindromic two-sided prime.
*881 is a minimal prime and a quartan prime.
*The number 919 is a cuban prime, and the largest known non-repunit palindromic permutable prime.
*929 is a palindromic Proth prime.
*977 is the third largest known Stern prime.
*The number 991 is a centered polygonal number, a minimal prime and the largest known non-repunit permutable prime.
*The number 1,093 is the smallest .
*1,187 and 1,493 are the two largest known Stern primes.
*1,597 is a and a pancake number.
*2,311 is a centered decagonal number and a prime .
*3,137 is a two-sided prime.
*The number 3,511 is the largest known Wieferich prime.
*3,797 is a two-sided prime.
*6,469 and 6,949 are minimal primes.
*7,919 is the 1,000th prime.
*9,001 is a minimal prime.
**It also refers to the "it’s over 9000" Internet meme.Cloudy's large number list Part 2 - Cloudy's googology
*9,049, 9,649 and 9,949 are minimal primes.
*14,197 is a pancake number and a Perrin prime.
*The number 16,843 is the smallest .Prime Curios!: 16843
*26,861 is the smallest prime for which the number of 4k + 1 primes exceed the number of 4k + 3 primes up to n.
*42,841 is a cuban prime and a star prime.
*60,649 is the only 5-digit minimal prime.
*\(65,537=2^{2^4}+1\) is the largest known Fermat prime.
*104,729 is the 10,000th prime.
*148,091 is the largest known number n for which both and are probable prime numbers.
*The number 262,657 is one of only four known , which are neither Fermat nor s.
*By fitting the least-degree polynomial to the first n odd primes, one can attempt to guess the (n + 1)-st odd prime, but this will give almost always incorrect results, which can be prime or composite, and positive or negative. The absolute value of the first negative prime obtained in this way is equal to 281,581.
**It is also the number of on Friday, January 27, 1984 in .Prime Curios!: 281581 Retrieved 2023-09-14.
*The number 294,001 is the smallest .
*666,649 is a minimal prime.
*The number 739,397 is the largest two-sided prime.
*946,669 is a minimal prime.
*The number 999,331 is the largest known non- .
*999,983 is the largest prime number smaller than 1,000,000; and, as such, the largest Class 1 number to be prime.
*1,000,003 is the smallest prime number larger than 1,000,000; and, as such, the smallest Class 2 number to be prime.
*1,299,709 is the 100,000th prime.
*The number 2,124,679 is the largest known Wolstenholme prime.
*The numbers 60,000,049, 66,000,049 and 66,600,049 are the three largest minimal primes.
*The number 15,485,863 is the 1,000,000th prime.
*The number 73,939,133 is the largest right-truncatable prime.
*The number 799,636,997 is the largest palindromic left-truncatable prime and the largest palindromic truncatable prime.
*The number 982,451,653 is the 50,000,000th prime number.The first fifty million primes. The Prime Pages.
*1,679,457,781 and 1,938,092,824,081 are cuban primes and star primes.
*The number 608,981,813,029 is the smallest prime for which the number of 3k + 1 primes exceed the number of 3k + 2 primes up to n.
*The number 4,432,676,798,593 is one of only four known Mersenne–Fermat primes, which are neither Fermat nor Mersenne primes.
*The number 29,996,224,275,833 is the 1,000,000,000,000th prime number.The Prime Database: The Nth Prime Page
*9,007,199,254,740,881 is a positive integer equal to \(2^{53} - 111\). It is notable in computer science for being the largest prime number which can be represented exactly in the double floating-point format (which has a 53-bit significand).
*The number 357,686,312,646,216,567,629,137 is the largest left-truncatable prime.
*10100+267 is the first prime after a googol. This number has been named as "gooprol".
*10999+7 is the smallest titanic prime.
*The number \(\frac{10^{1,031}-1}{9}\) is the the largest known base 10 repunit prime.
*The number \(\frac{2^{3,481}-1}{2^{59}-1}\) is one of only four known that are neither nor s.
**Both of the last two primes have 1,031 digits, and start with the digit “1”.
*The number 1010,006+941,992,101×104,999+1 is the largest known emirp.
*The number 2,618,163,402,417×21,290,000-1 is the largest known Sophie Germain prime.
*The numbers 2,996,863,034,895×21,290,000±1 are the largest known twin primes.
*The number 282,589,933-1 is the largest known prime as of December 2020.
== Decimal expansions ==
For \(\frac{2^{3,481}-1}{2^{59}-1}\):
== See also ==
*Mersenne-related numbers
*Numbers in number theory
*Mills' constant
*Wright's primes
== Sources ==
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Category:Numbers
Category:Lists
Category:Class 1
Category:Unnamed numbers with 4 to 6 digits
Category:Numbers with 3 digits
Category:Class 2
Category:Class 3
Category:Unnamed numbers with 7 to 29 digits
Category:Numbers with 1000 to 1000000 digits
Category:Lower class 3