A062326 Primes p such that p^2 - 2 is also prime.

2, 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89, 103, 107, 127, 131, 139, 173, 191, 211, 223, 233, 239, 257, 293, 313, 337, 359, 421, 443, 449, 467, 491, 523, 541, 569, 587, 607, 653, 677, 719, 727, 733, 743, 751, 757, 761, 797, 811, 863, 881, 1013, 1021

Offset: 1

Author

Reiner Martin, Jul 12 2001

Keywords

• nonn
• nice

Comments

When p and p^2 - 2 are both prime, the fundamental solution of the Pell equation x^2 - n*y^2 = 1, for n = p^2 - 2, is x = p^2 - 1 and y = p. See A109748 for the case of n and x both prime. - T. D. Noe, May 19 2007

3 is the only prime p such that p^2 + 2 and p^2 - 2 are both primes. - Jaroslav Krizek, Nov 25 2013 (note that p^2 + 2 is composite for all primes p >= 5. - Joerg Arndt, Jan 10 2015)

For all primes p except for p = 3, p^2 + 2 is multiple of 3 (see A061725). - Zak Seidov, Feb 19 2015

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A049002 (p^2-2).

Cf. A137291, A010051, A004901, A000040.

Programs

Haskell

import Data.List (elemIndices)
a062326 = a000040 . a137291
a062326_list = map (a000040 . (+ 1)) $
               elemIndices 1 $ map a010051' a049001_list
-- Reinhard Zumkeller, Jul 30 2015

Magma

[ p: p in PrimesUpTo(1100) | IsPrime(p^2-2) ]; // Klaus Brockhaus, Jan 01 2009

Mathematica

Select[Prime[Range[500]], PrimeQ[#^2 - 2] &] (* Harvey P. Dale, Sep 20 2011 *)

PARI

{ n=0; forprime (p=2, 5*10^5, if (isprime(p^2 - 2), write("b062326.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009