A360155 Primes of the form m^2 + 2*(k+1)^2 such that m^2 + 2*k^2 is also prime.

17, 59, 89, 131, 137, 233, 401, 449, 587, 617, 659, 683, 857, 971, 1019, 1097, 1217, 1283, 1361, 1481, 1499, 1571, 1667, 1787, 1889, 2081, 2129, 2411, 2441, 2531, 2729, 2843, 2969, 3137, 3203, 3257, 3371, 3491, 3617, 4019, 4073

Offset: 1

Author

Ludovic Schwob, Jan 28 2023

Keywords

• nonn

Comments

Primes of the form m^2 + 2*k^2 are the norms of prime elements of Z[i*sqrt(2)]. Pairs of primes of the form (m^2 + 2*k^2, m^2 + 2*(k+1)^2) correspond to primes in Z[i*sqrt(2)] differing by i*sqrt(2).

A prime cannot simultaneously be the lesser of such a pair and the greater of another.

Examples

			The first 3 such prime pairs are
  (11,17) = (3^2 + 2*1^2, 3^2 + 2*2^2) with m=3 and k=1,
  (41,59) = (3^2 + 2*4^2, 3^2 + 2*5^2) with m=3 and k=4,
  (83,89) = (9^2 + 2*1^2, 9^2 + 2*2^2) with m=9 and k=1.
		

Crossrefs

See A360154 for lesser primes.

Cf. A000040 (prime numbers).

Cf. A033203 (primes of the form m^2 + 2*k^2).

Formula

If m^2 + 2*k^2 and m^2 + 2*(k+1)^2 are primes, then m == 3 (mod 6) and k == 1 (mod 3).