A066436 Primes of the form 2*n^2 - 1.

7, 17, 31, 71, 97, 127, 199, 241, 337, 449, 577, 647, 881, 967, 1151, 1249, 1567, 2311, 2591, 2887, 3041, 3361, 3527, 3697, 4049, 4231, 4801, 4999, 5407, 6271, 6961, 7687, 7937, 8191, 9521, 10657, 11551, 12799, 13121, 14449, 15137, 16561

Offset: 1

Author

N. J. A. Sloane, Jan 09 2002

Keywords

• nonn

Comments

It is conjectured that this sequence is infinite.

Also primes p such that 8p + 8 is a square. - Cino Hilliard, Dec 18 2003

Also primes p such that 2p+2 is square; also primes p such that (p+1)/2 is square. - Ray Chandler, Sep 15 2005

Arithmetic numbers which are squares, A003601(p)=A000290(k), p prime, k integer. sigma_1(p)/sigma_0(p)=k^2; p prime, k integer. - Ctibor O. Zizka, Jul 14 2008

References

• D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 31.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Project Euler, Problem 216: Investigating the primality of numbers of the form 2n^2-1.

Crossrefs

See A066049 for the values of n, see A091176 for prime index.

Cf. A090697, A110558, A003601, A000290.

Programs

Magma

[ p: n in [1..100] | IsPrime(p) where p is 2*n^2-1 ]; // Klaus Brockhaus, Dec 29 2008

Mathematica

Select[2*Range[200]^2-1,PrimeQ] (* Harvey P. Dale, Aug 29 2016 *)

PARI

{ n=0; for (m=1, 10^9, p=2*m^2 - 1; if (isprime(p), write("b066436.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 14 2010