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.
 * 5 is the second Fermat prime.
 * 7 is the second Mersenne prime.
 * 17 is the third Fermat prime.
 * 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 palindromic prime.
 * 113 is a number following 112 and preceding 114. This number is prime.
 * 113 is also the 11th.
 * 163 is the largest.
 * 227 is the fourth largest known.
 * It is also the number of pips in a set.
 * The number 257 is a \(2^{2^3}+1\).
 * The isotope -257 is the heaviest that can be formed by  from naturally occuring.
 * 541 is the 100th prime.
 * The number 563 is the largest known.
 * 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, ….
 * 719 is a prime number. As 119, 121 and 721 are all composite, it is the only 3-digit factorial prime.
 * It is also the number of hours in a 30-day month (April, June, September or November) containing a spring transition.
 * 977 is the third largest known Stern prime.
 * The number 1,093 is the smallest.
 * 1,187 and 1,493 are the two largest known Stern primes.
 * The number 3,511 is the largest known Wieferich prime.
 * 7,919 is the 1,000th prime.
 * The number 16,843 is the smallest.
 * \(65,537=2^{2^4}+1\) is the largest known Fermat 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.
 * 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.
 * The number 2,124,679 is the largest known Wolstenholme prime.
 * The number 982,451,653 is the 50,000,000th prime number.
 * 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.
 * 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  floating-point format (which has a 53-bit significand).
 * 10100+267 is the first prime after a googol. This number has been named as "gooprol".
 * 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 277,232,917-1 is the largest known prime.

Decimal expansions
For \(\frac{2^{3,481}-1}{2^{59}-1}\):

13324323309828620642589590565533923081483782150370704217672886885162756499559745515820505025366633291782689824970508202932981177480858933989443161914437860223829486481498201271806160710212419319981647591766471221549778791249081428838239687282350328447116067333733212653644768614482418519392989453221962115799024522405104498901459713737808685662443413595655349375239048341550958241450638814760944590236658374229179290977642222726256754317985049014925694253475958911625949983248927943732325461584057736439218050753700932773508299940797760652182226128976123104989251256067036990378850337686156071082494239176664863922935977210027442841831990203885909738423666863072782748227328328682294854519033727328136521782531700308411697804383954107548151069972793277926158786752065849297036891260767326465784518800758457811377420171439984071715181951117763117140248357060929148011779659503510742142318403354432945158174149891228101860550295710148830648133189336855311682287121680507578718514924229191266447187940645267114826510722293606459113473