A007641 Primes of the form 2*k^2 + 29.

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

Author

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

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A005846, A050265, A352800.

Programs

Magma

[a: n in [0..60] | IsPrime(a) where a is 2*n^2+29]; // Vincenzo Librandi, Mar 20 2013

Mathematica

Select[Table[2 n^2 + 29, {n, 0, 70}], PrimeQ] (* 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

Extensions

Edited by Erich Friedman, Feb 09 2002