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

1,1

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.

Links

Table of n, a(n) for n=1..41.

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.

Example

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).

Sequence in context: A031389 A178495 A309254 * A217624 A097991 A158187

Adjacent sequences: A360151 A360152 A360153 * A360155 A360156 A360157

Keyword

nonn

Author

Ludovic Schwob, Jan 28 2023

Status

approved