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

%I #23 Feb 18 2023 20:50:37

%S 17,59,89,131,137,233,401,449,587,617,659,683,857,971,1019,1097,1217,

%T 1283,1361,1481,1499,1571,1667,1787,1889,2081,2129,2411,2441,2531,

%U 2729,2843,2969,3137,3203,3257,3371,3491,3617,4019,4073

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

%C 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).

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

%F 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).

%e The first 3 such prime pairs are

%e (11,17) = (3^2 + 2*1^2, 3^2 + 2*2^2) with m=3 and k=1,

%e (41,59) = (3^2 + 2*4^2, 3^2 + 2*5^2) with m=3 and k=4,

%e (83,89) = (9^2 + 2*1^2, 9^2 + 2*2^2) with m=9 and k=1.

%Y See A360154 for lesser primes.

%Y Cf. A000040 (prime numbers).

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

%K nonn

%O 1,1

%A _Ludovic Schwob_, Jan 28 2023