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%I #22 Apr 19 2019 13:45:10
%S 113,257,337,353,577,593,881,1153,1201,1217,1249,1553,1601,1777,1889,
%T 2113,2129,2273,2593,2657,2689,2833,3089,3121,3137,3217,3313,3361,
%U 3761,4001,4049,4177,4273,4289,4481,4513,4657,4721,4801,4817,4993,5233,5297,5393
%N Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32*y^2 and x^2 + 64*y^2.
%C Kaplansky showed that prime numbers congruent to 1 modulo 16 are representable by both or neither of the quadratic forms x^2 + 32*y^2 and x^2 + 64*y^2. This sequence corresponds to those representable by both, and A325068 corresponds to those representable by neither.
%C Also, Kaplansky showed that prime numbers congruent to 9 modulo 16 are representable by exactly one of these quadratic forms. A325069 corresponds to those representable by the first form and A325070 to those representable by the second form.
%C Brink provided similar results for other congruences.
%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="/A325067/a325067.gp.txt">PARI program for A325067</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 1201:
%e - 1201 is a prime number,
%e - 1201 = 75*16 + 1,
%e - 1201 = 7^2 + 32*6^2 = 25^2 + 64*3^2,
%e - hence 1201 belongs to the sequence.
%o (PARI) See Links section.
%Y Cf. A094407, A105126, A325068, A325069, A325070.
%Y See A325071, A325072, A325073 and A325074 for similar results in congruences modulo 16.
%Y See A325075, A325076, A325077 and A325078 for similar results in congruences modulo 39.
%Y See A325079, A325080, A325081 and A325082 for similar results in congruences modulo 55.
%Y See A325083, A325084, A325085 and A325086 for similar results in congruences modulo 112.
%Y See A325087, A325088, A325089 and A325090 for similar results in congruences modulo 240.
%K nonn
%O 1,1
%A _Rémy Sigrist_, Mar 27 2019
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