A prime number 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.
  • 17 is the third Fermat prime and the third Stern 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 centered decagonal number and a palindromic prime.
  • 113 is a Sophie Germain prime.[1]
    • 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 Milü. It's equal to 3.141592920...
  • 131 is a palindromic permutable prime.
  • 137 is a centered polygonal number and the fifth largest known Stern prime.
    • It is also approximately the reciprocal of the fine-structure constant.
    • Since 177 and 183 kHz 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 Psalms contains 151 psalms.
    • Finally, it is also the number of species in the first Pokémon generation.
  • 163 is the largest Heegner number. For this reason, it has been used in the Ramanujan constant.
  • The emirp 167 is the first number n, such that 4n is larger than a googol.
    • It is also the number of hours in the spring DST transition week.
    • And in China, the Band III starts at 167 MHz.
  • 181 is a centered square number, a centered pentagonal number, a palindromic prime and a star prime.
  • 191 is a centered polygonal number, a palindromic prime and a Thabit prime.
    • It is also the number of non-control ISO/IEC 8859 characters.
  • 193 is a cuban prime, a Pierpont prime, a Proth prime, and the number of ways to add seven ordinals.
  • 199 is a centered triangular number, a Lucas prime and a permutable prime.
  • 211 is a centered decagonal number, a centered polygonal number and a prime Euclid number.
  • 227 is the fourth largest known Stern prime.
  • 239 is a Sophie Germain prime. It was the PEGG value on May 22nd, 2017.
    • Since 2392 + 1 = 2 × 134, it also appears in many Machin-like formulae.
    • It is the largest integer which cannot be written as a sum of eight nonnegative cubes; 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 Fermat prime \(2^{2^3}+1\).
  • The number 271 is a cuban prime, and the only prime house number.
  • 277 is a centered polygonal number and a Perrin prime.
  • 311 is a permutable prime and a right-truncatable prime.
  • 313 is a centered square number and a palindromic prime.
  • 317 is an index of a repunit prime and a two-sided prime.
  • 337 is a left-truncatable prime, a permutable prime and a quartan prime.
  • 353 is a palindromic Proth prime.
  • 373 is a palindromic permutable prime.
  • 383 is a palindromic prime, a Thabit prime and a Woodall prime.
  • 409 is a minimal prime.
    • Since it is the integral part of the quotient of the DVD sector size (2 KiB) by the 40-bit key size, it is also the number of keys in the CSS 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 Wilson prime.
  • 613 is a centered square number and a left-truncatable prime.
  • 619 is an alternating factorial prime and a strobogrammatic prime.
  • 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 Fermat or Miller-Rabin, and the primality by Pratt. The resulting sequence starts with 1, 2, 3, 7, 29, 59, 709, … (OEIS A061092).
    • 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, … (OEIS A007097).
    • Furthermore, it is the largest number n, for which en can be represented in the double-precision floating-point format.
    • Finally, it is the number of seats in the 19th Bundestag, 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 factorial prime.
    • It is also the number of hours in a 30-day month (April, June, September or November) containing a spring DST transition.
  • 727 is a palindromic prime.
  • 733 is a permutable prime and a right-truncatable prime.
  • 757 is a Hogben number and a palindromic prime.
  • 773 is a tetranacci number 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 Wieferich prime.
  • 1,187 and 1,493 are the two largest known Stern primes.
  • 1,597 is a Fibonacci prime and a pancake number.
  • 2,311 is a centered decagonal number and a prime Euclid number.
  • 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.[2]
  • 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 Wolstenholme prime.[3][4]
  • 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.
  • 148,091 is the largest known number n for which both F(n) and L(n) are probable prime numbers.
  • The number 262,657 is one of only four known Mersenne–Fermat primes, which are neither Fermat nor Mersenne primes.
  • 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.[5]
  • The number 294,001 is the smallest weakly prime.
  • 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-repunit circular prime.
  • 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.[3][4]
  • The numbers 60,000,049, 66,000,049 and 66,600,049 are the three largest minimal primes.
  • 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.[7]
  • 1,679,457,781 and 1,938,092,824,081 are cuban primes and star primes.
  • 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.[8]
  • 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 Mersenne–Fermat primes that are neither Fermat nor Mersenne primes.
    • 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 \(\frac{2^{3,481}-1}{2^{59}-1}\):

13324323309828620642589590565533923081483782150370704217672886885162756499559745515820505025366633291782689824970508202932981177480858933989443161914437860223829486481498201271806160710212419319981647591766471221549778791249081428838239687282350328447116067333733212653644768614482418519392989453221962115799024522405104498901459713737808685662443413595655349375239048341550958241450638814760944590236658374229179290977642222726256754317985049014925694253475958911625949983248927943732325461584057736439218050753700932773508299940797760652182226128976123104989251256067036990378850337686156071082494239176664863922935977210027442841831990203885909738423666863072782748227328328682294854519033727328136521782531700308411697804383954107548151069972793277926158786752065849297036891260767326465784518800758457811377420171439984071715181951117763117140248357060929148011779659503510742142318403354432945158174149891228101860550295710148830648133189336855311682287121680507578718514924229191266447187940645267114826510722293606459113473

See also

Sources

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