A130229 - OEIS

A130229

Primes p == 5 (mod 8) such that the Diophantine equation x^2 - p*y^2 = -4 has no solution in odd integers x, y.

%I #28 Sep 05 2024 16:08:48

%S 37,101,197,269,349,373,389,557,677,701,709,757,829,877,997,1213,1301,

%T 1613,1861,1901,1949,1973,2069,2221,2269,2341,2357,2621,2797,2837,

%U 2917,3109,3181,3301,3413,3709,3797,3821,3853,3877,4013,4021,4093

%N Primes p == 5 (mod 8) such that the Diophantine equation x^2 - p*y^2 = -4 has no solution in odd integers x, y.

%C For the Diophantine equation x^2 - p*y^2 = -4 to have a solution in odd integers x, y we must have p == 5 (mod 8).

%C Calculated using Dario Alpern's quadratic Diophantine solver, see link.

%C Suggested by a discussion on the Number Theory Mailing List, circa Aug 01 2007.

%H Robin Visser, <a href="/A130229/b130229.txt">Table of n, a(n) for n = 1..10000</a>

%H Dario Alpern, <a href="https://www.alpertron.com.ar/QUAD.HTM">Generic two integer variable equation solver</a>.

%H Florian Breuer, <a href="https://carmamaths.org/breuer/DucciPell_baustms_v2.pdf">Periods of Ducci sequences and odd solutions to a Pellian equation</a>, University of Newcastle, Australia, 2018.

%H Florian Breuer and Cameron Shaw-Carmody, <a href="https://carmamaths.org/breuer/ParityBias-v1.pdf">Parity bias in fundamental units of real quadratic fields</a>, Univ. Newcastle (Australia), Comp.-Assisted Res. Math. Appl. (2024). See pp. 1-4.

%H J. Xue, T.-C. Yang, C.-F. Yu, <a href="https://arxiv.org/abs/1404.2978">Supersingular abelian surfaces and Eichler class number formula</a>, arXiv preprint arXiv:1404.2978, 2014

%H Jiangwei Xue, TC Yang, CF Yu, <a href="https://arxiv.org/abs/1603.02789">Numerical Invariants of Totally Imaginary Quadratic Z[sqrt{p}]-orders</a>, arXiv preprint arXiv:1603.02789, 2016

%Y Cf. A130230.

%K nonn

%O 1,1

%A _Warut Roonguthai_, Aug 06 2007