See also the excellent page of Chris Caldwell.

  • The largest known prime is 257885161-1 (17,425,170 digits), found by Curtis Cooper within GIMPS on January 25, 2013.
  • The largest known twin primes are 33218925*2^169690+/-1 (51090 digits), found by Papp and Gallot in 2002. See also Chris Caldwell's Prime Pages which may be more up-to-date.
  • The largest known arithmetic progression of primes contains 24 primes: 468395662504823 + k*45872132836530, with k=0..23: it was discovered on January 18 2007 by Jaroslaw Wroblewski. The previous record had 23 primes: 56211383760397 + k*44546738095860, with k=0..22; it was discovered on 24 July 2004 by Markus Frind, Paul Jobling and Paul Underwood. Green and Tao published in April 2004 a preprint with a proof that for any given k ≥ 3, there exist arithmetic progressions of primes of length k.
  • The longest known sequence of consecutive primes in arithmetic progression has 10 primes, and was found in March 1998 by Manfred Toplic, one of about 100 contributors with about 200 machines. See also this site and this one.
  • The only known Wieferich primes are 1093 and 3511. Wieferich primes are primes such that 2^(p-1) = 1 mod p^2. Richard Crandall, Karl Dilcher and Carl Pomerance looked for Wieferich primes up to 4*10^12, and found no other (Math. of Comp. 217, 1997). They kept a table of "special instances" i.e. numbers p such that 2^((p-1)/2) (mod p^2) = +/-1+A*p with |A| less or equal to 100. Here is a file containing some pairs (p,q) of primes such that p^(q-1) = 1 mod (q^2). For each p, the value qmax indicates the upper bound of the search. These are known as Fermat's quotients. Wieferich primes correspond to the case p=2. Richard McIntosh is searching up to 8 trillion, and Rich Brown from 8 through 10 trillion.
  • The only known Wilson primes are 5, 13 and 563. A Wilson prime is a prime number such that (p-1)! = -1 mod p^2. These are the only known Wilson primes up to 5*10^8 (Richard Crandall, Karl Dilcher and Carl Pomerance, Math. of Comp. 217, 1997). See also The Book of Prime Number Records, P. Ribenboim, Springer, 1989. See the table of special instances.
  • The largest known value of the function pi(x) is pi(10^21) = 21 127 269 486 018 731 928, obtained by Xavier Gourdon on October 27, 2000. The previous record was pi(10^20)=2,220,819,602,560,918,840, obtained by Marc Deleglise and Paul Zimmermann in 13 days of cpu time on a DEC-ALPHA 5/250 and checked on a R8000, using a program written together with Joel Rivat, who had already computed pi(10^18)=24739954287740860 (Math. of Comp. 65, 1996).
  • The largest known Cullen and Woodall primes are C[481899] (145072 digits, discovered by Masakatu Morri on September 30, 1998) and W[98726] (29725 digits) by Jeffrey Young in 1997 where C[n]=n*2^n+1 and W[n]=n*2^n-1. See "New Cullen Primes" by Wilfrid Keller, Math. of Comp. vol. 64 nb. 212, oct. 1995, pages 1733-1741. Wilfrid Keller and Paul Leyland also keep tables of the factorizations of Cullen and Woodall numbers.
  • The largest known Sophie Germain prime is 1213822389*2^81131-1 (24432 digits), found by Michael Angel, Dirk Augustin and Paul Jobling in August, 2002. Sophie Germain primes P are such that P and 2P+1 are prime. See the paper from Harvey Dubner in Math. of Comp. v65 n213, 1996, 393-396.
  • The largest known candidate repunit prime is R(49081)=(10^49081-1)/9, found by Harvey Dubner on September 9, 1999 [it is only candidate since it has not been really proved prime]. The only known repunit primes are R(2), R(19), R(23), R(317), R(1031).
  • The longest known Cunningham chain of the 2nd kind is of length 16, found by Tony Forbes on 5 December 1997. It begins with 3203000719597029781, and further primes are obtained by iterating 2*p-1 (a Cunningham chain can also use 2*p+1). See Tony's announcement for more details, and A057330.
  • Prime gaps. Thomas R. Nicely and Bertil Nyman found the first occurrence of prime gap of 1000 or greater, namely the gap of 1132 following the prime 1693182318746371. Harvey Dubner found two gaps of length more than 2000 starting with 51-digit numbers, one gap of 12540 near 10384, and one gap of at least 50206 near 3 × 101883.