|
%I #15 Apr 12 2019 18:54:18
%S 113,193,337,401,641,1009,1201,1297,2689,2801,3089,3137,3217,3329,
%T 3361,3761,3889,4337,4481,5009,5153,5233,5441,5569,6113,6337,6353,
%U 6449,6577,6673,7681,7841,8513,8737,8929,9041,9137,9521,9601,9697,10369,10529,10753
%N Prime numbers congruent to 1, 65 or 81 modulo 112 neither representable by x^2 + 14*y^2 nor by x^2 + 448*y^2.
%C Brink showed that prime numbers congruent to 1, 65 or 81 modulo 112 are representable by both or neither of the quadratic forms x^2 + 14*y^2 and x^2 + 448*y^2. A325083 corresponds to those representable by both, and this sequence 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="/A325084/a325084.gp.txt">PARI program for A325084</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 113:
%e - 113 is a prime number,
%e - 113 = 1*112 + 1,
%e - 113 is neither representable by x^2 + 14*y^2 nor by x^2 + 448*y^2,
%e - hence 113 belongs to this sequence.
%o (PARI) See Links section.
%Y See A325067 for similar results.
%Y Cf. A325083.
%K nonn
%O 1,1
%A _Rémy Sigrist_, Mar 28 2019
|
