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Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + x*y + 127*y^2.
3

%I #13 Apr 12 2019 18:53:24

%S 127,199,283,337,433,571,727,829,883,907,1213,1291,1297,1447,1531,

%T 1609,1663,1741,2053,2383,2389,2677,3169,3301,3319,3631,3691,3709,

%U 3769,3793,4003,4099,4159,4549,4567,4651,4729,4801,4957,5347,5407,5431,5563,5821,6133

%N Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + x*y + 127*y^2.

%C Brink showed that prime numbers congruent to 4, 10 or 25 modulo 39 are representable by exactly one of the quadratic forms x^2 + x*y + 10*y^2 or x^2 + x*y + 127*y^2. A325077 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

%H David Brink, <a href="https://doi.org/10.1016/j.jnt.2008.04.007">Five peculiar theorems on simultaneous representation of primes by quadratic forms</a>, Journal of Number Theory 129(2) (2009), 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.

%H Rémy Sigrist, <a href="/A325078/a325078.gp.txt">PARI program for A325078</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kaplansky%27s_theorem_on_quadratic_forms">Kaplansky's theorem on quadratic forms</a>

%e Regarding 127:

%e - 127 is a prime number,

%e - 127 = 3*39 + 10,

%e - 127 = 0^2 + 0*1 + 127*1^2,

%e - hence 127 belongs to this sequence.

%o (PARI) See Links section.

%Y See A325067 for similar results.

%Y Cf. A325077.

%K nonn

%O 1,1

%A _Rémy Sigrist_, Mar 28 2019