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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.
1
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.
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
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Jan 28 2023
STATUS
approved