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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

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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.