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A360154 Primes of the form m^2 + 2*k^2 such that m^2 + 2*(k+1)^2 is also prime. 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Jan 28 2023
STATUS
approved

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Last modified July 23 09:37 EDT 2024. Contains 374547 sequences. (Running on oeis4.)