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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..45.

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.

Rémy Sigrist, PARI program for A325075

Wikipedia, Kaplansky's theorem on quadratic forms

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: A047652 A308788 A308796 * A020357 A050967 A071382

Adjacent sequences:  A325072 A325073 A325074 * A325076 A325077 A325078

KEYWORD

nonn

AUTHOR

Rémy Sigrist, Mar 28 2019

STATUS

approved

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Last modified February 21 02:02 EST 2020. Contains 332086 sequences. (Running on oeis4.)