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A325075
Prime numbers congruent to 1, 16 or 22 modulo 39 representable by both x^2 + x*y + 10*y^2 and x^2 + x*y + 127*y^2.
3
139, 157, 367, 523, 547, 607, 991, 997, 1153, 1171, 1231, 1249, 1381, 1459, 1483, 1693, 1933, 1951, 2011, 2029, 2473, 2557, 3121, 3181, 3253, 3259, 3433, 3511, 3643, 3877, 4111, 4447, 4603, 4663, 4759, 5521, 5749, 5827, 6007, 6163, 6217, 6301, 6397, 6451, 6553
OFFSET
1,1
COMMENTS
Brink showed that prime numbers congruent to 1, 16 or 22 modulo 39 are representable by both or neither of the quadratic forms x^2 + x*y + 10*y^2 and x^2 + x*y + 127*y^2. This sequence corresponds to those representable by both, and A325076 corresponds to those representable by neither.
LINKS
David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129(2) (2009), 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.
EXAMPLE
Regarding 997:
- 997 is a prime number,
- 997 = 25*39 + 22,
- 997 = 27^2 + 27*4 + 10*4^2 = 29^2 + 29*1 + 127*1^2,
- hence 997 belongs to this sequence.
PROG
(PARI) See Links section.
CROSSREFS
See A325067 for similar results.
Cf. A325076.
Sequence in context: A308788 A308796 A334564 * A020357 A050967 A071382
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Mar 28 2019
STATUS
approved