A360154 Primes of the form m^2 + 2*k^2 such that m^2 + 2*(k+1)^2 is also prime.

11, 41, 83, 107, 113, 227, 347, 443, 521, 563, 593, 641, 827, 929, 953, 1091, 1187, 1193, 1259, 1409, 1427, 1553, 1601, 1697, 1811, 1979, 2003, 2297, 2339, 2393, 2699, 2801, 2819, 3011, 3089, 3209, 3251, 3449, 3467, 3929, 3947

Offset: 1

Author

Ludovic Schwob, Jan 28 2023

Keywords

• nonn

Comments

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

A prime cannot be simultaneously the lesser of one such couple and the greater of another.

Examples

			The first 3 prime couples of the form (m^2 + 2*k^2, m^2 + 2*(k+1)^2) are (11,17) = (3^2 + 2*1^2, 3^2 + 2*2^2), (41,59) = (3^2 + 2*4^2, 3^2 + 2*5^2) and (83,89) = (9^2 + 2*1^2, 9^2 + 2*2^2).
		

Crossrefs

See A360155 for greater values.

Cf. A000040 (prime numbers).

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

Formula

If (m^2 + 2*k^2, m^2 + 2*(k+1)^2) is a prime couple, then m is congruent to 3 modulo 6 and k is congruent to 1 modulo 3.