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A prime number is an integer greater than 1 that has no divisors other than 1 and itself.

List of notable prime numbers Edit

  • 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.
  • 163 is the largest Heegner number.
  • The number 257 is a Fermat prime \(2^{2^3}+1\).
  • 541 is the 100th prime.
  • The number 563 is the largest known Wilson prime.
  • 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).
  • 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.
  • The number 1,093 is the smallest Wieferich prime.
  • The number 3,511 is the largest known Wieferich prime.
  • 7,919 is the 1,000th prime.
  • The number 16,843 is the smallest Wolstenholme 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.[2]
  • 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.[4]
  • 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.[5]
  • 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).
  • 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 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 277,232,917-1 is the largest known prime.

Decimal expansions Edit

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

13324323309828620642589590565533923081483782150370704217672886885162756499559745515820505025366633291782689824970508202932981177480858933989443161914437860223829486481498201271806160710212419319981647591766471221549778791249081428838239687282350328447116067333733212653644768614482418519392989453221962115799024522405104498901459713737808685662443413595655349375239048341550958241450638814760944590236658374229179290977642222726256754317985049014925694253475958911625949983248927943732325461584057736439218050753700932773508299940797760652182226128976123104989251256067036990378850337686156071082494239176664863922935977210027442841831990203885909738423666863072782748227328328682294854519033727328136521782531700308411697804383954107548151069972793277926158786752065849297036891260767326465784518800758457811377420171439984071715181951117763117140248357060929148011779659503510742142318403354432945158174149891228101860550295710148830648133189336855311682287121680507578718514924229191266447187940645267114826510722293606459113473

See also Edit

Sources Edit

  1. https://en.wikipedia.org/wiki/113_(number)
  2. OEIS A140118
  3. http://primes.utm.edu/~caldwell/curios/page.php/281581.html
  4. http://primes.utm.edu/lists/small/millions/
  5. http://primes.utm.edu/nthprime/