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

1,1

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.

Links

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

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).

Example

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).

Sequence in context: A095089 A106922 A110092 * A141896 A104165 A031391

Adjacent sequences: A360152 A360153 A360154 * A360156 A360157 A360158

Keyword

nonn

Author

Ludovic Schwob, Jan 28 2023

Status

approved