|
%I #15 Apr 12 2019 18:53:31
%S 71,251,311,631,661,691,751,881,1061,1171,1181,1321,1571,1721,1741,
%T 1901,1951,2341,2531,2621,2671,2711,2731,2971,3191,3271,3371,3491,
%U 3631,3701,3851,3881,4481,4591,4651,5261,5471,5501,5531,5581,5641,5701,5861,6121,6271
%N Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 representable by both x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2.
%C Brink showed that prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 are representable by both or neither of the quadratic forms x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2. This sequence corresponds to those representable by both, and A325080 corresponds to those representable by neither.
%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="/A325079/a325079.gp.txt">PARI program for A325079</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 881:
%e - 881 is a prime number,
%e - 881 = 16*55 + 1,
%e - 881 = 3^2 + 3*(-8) + 14*(-8)^2 = 28^2 + 28*1 + 69*1^2,
%e - hence 881 belongs to this sequence.
%o (PARI) See Links section.
%Y See A325067 for similar results.
%Y Cf. A325080.
%K nonn
%O 1,1
%A _Rémy Sigrist_, Mar 28 2019
|
