A204189 Benoît Perichon's 26 primes in arithmetic progression.

43142746595714191, 48425980631694091, 53709214667673991, 58992448703653891, 64275682739633791, 69558916775613691, 74842150811593591, 80125384847573491, 85408618883553391, 90691852919533291, 95975086955513191, 101258320991493091, 106541555027472991, 111824789063452891, 117108023099432791, 122391257135412691, 127674491171392591, 132957725207372491, 138240959243352391, 143524193279332291, 148807427315312191, 154090661351292091, 159373895387271991, 164657129423251891, 169940363459231791, 175223597495211691

Offset: 1

Author

Jonathan Sondow, Jan 14 2012

Keywords

• nonn
• fini
• full
• easy

Comments

Longest known arithmetic progression of primes as of Jan 14, 2012.

Discovered on Apr 12 2010 by Benoît Perichon using software by Jaroslaw Wroblewski and Geoff Reynolds in a distributed PrimeGrid project.

References

• R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag, 1994, A5 and A6.
• P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1989, p. 224.

Links

• J. K. Andersen, Primes in Arithmetic Progression Records.
• T. Eisner and R. Nagel, Arithmetic progressions-an operator theoretic view, Discrete and continuous dynamical systems series S, Volume 6, Number 3, June 2013 pp. 657-667; doi:10.3934/dcdss.2013.6.657. - From _N. J. A. Sloane_, Feb 03 2013
• A. Granville, Prime Number Patterns, Amer. Math. Monthly 115 (2008), 279-296.
• B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167 (2008), 481-547.
• PrimeGrid, AP26 Search.
• Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
• Wikipedia, Primes in arithmetic progression.
• J. Wroblewski, How to search for 26 primes in arithmetic progression?, May 23, 2008.
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Cf. A002110, A033188, A033189, A033290, A260751, A261140, A327760, A363980, A374949.

Programs

Mathematica

a[1] := 43142746595714191; a[n_] := a[n] = a[n - 1] + 5283234035979900; Table[a[n], {n, 26}] (* Alonso del Arte, Jan 14 2012 *)
Range[ 43142746595714191, 175223597495211691, 5283234035979900] (* Michael Somos, Jan 15 2012 *)

PARI

a(n)=5283234035979900*n+37859512559734291 \\ Charles R Greathouse IV, Jan 15 2012

Formula

a(n) = 43142746595714191 + 5283234035979900*(n-1) for n = 1, 2, ..., 26.

a(n) = 43142746595714191 + 23681770*23#*(n-1) for n = 1..26, where 23# = 2*3*5*7*11*13*17*19*23 = 223092870 = A002110(9).