A007641 Primes of the form 2*k^2 + 29. (Formerly M5219)

29, 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 1597, 1951, 2207, 2341, 2621, 2767, 2917, 3229, 3391, 3557, 3727, 4079, 4261, 4447, 4637, 4831, 5231, 5437

Offset

1,1

Comments

The first 29 terms of 2*k^2 + 29 (k = 0 to 28) are primes. This was discovered by Adrien-Marie Legendre. The sequence and its first 8 terms appear in the novel Code to Zero by Ken Follett. - Amiram Eldar, Apr 08 2017

Let P(k) = 2*k^2 + 29. The polynomial P(2*k - 28) = 8*k^2 - 224*k + 1597 produces prime values (not distinct) for k = 0 to 28. The polynomial P(3*k - 55) = 18*k^2 - 660*k + 6079 produces distinct prime values for k = 0 to 27. Cf. A050265. - Peter Bala, Apr 16 2018

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ken Follett, Code to Zero, New York: Signet, 2001, p. 18.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Adrien-Marie Legendre, Essai sur la théorie des nombres, Paris: Duprat, 1798, p. 10.

Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.

Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Mathematica

Select[Table[2 n^2 + 29, {n, 0, 70}], PrimeQ] (* Vincenzo Librandi, Mar 20 2013 *)

Prog

(Magma) [a: n in [0..60] | IsPrime(a) where a is 2*n^2+29]; // Vincenzo Librandi, Mar 20 2013
(PARI) list(lim)=my(v=List(), t); for(n=0, sqrtint((lim-29)\2), if(isprime(t=2*n^2+29), listput(v, t))); Vec(v) \\ Charles R Greathouse IV, Jan 20 2022

Crossrefs

Cf. A005846, A050265, A352800.

Sequence in context: A031062 A054056 A108112 * A050656 A050667 A288879

Adjacent sequences: A007638 A007639 A007640 * A007642 A007643 A007644

Keyword

nonn,easy

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Extensions

Edited by Erich Friedman, Feb 09 2002

Status

approved