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.

37, 101, 197, 269, 349, 373, 389, 557, 677, 701, 709, 757, 829, 877, 997, 1213, 1301, 1613, 1861, 1901, 1949, 1973, 2069, 2221, 2269, 2341, 2357, 2621, 2797, 2837, 2917, 3109, 3181, 3301, 3413, 3709, 3797, 3821, 3853, 3877, 4013, 4021, 4093

Offset: 1

Author

Warut Roonguthai, Aug 06 2007

Keywords

• nonn

Comments

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).

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

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

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• Dario Alpern, Generic two integer variable equation solver.
• Florian Breuer, Periods of Ducci sequences and odd solutions to a Pellian equation, University of Newcastle, Australia, 2018.
• Florian Breuer and Cameron Shaw-Carmody, Parity bias in fundamental units of real quadratic fields, Univ. Newcastle (Australia), Comp.-Assisted Res. Math. Appl. (2024). See pp. 1-4.
• J. Xue, T.-C. Yang, C.-F. Yu, Supersingular abelian surfaces and Eichler class number formula, arXiv preprint arXiv:1404.2978, 2014
• Jiangwei Xue, TC Yang, CF Yu, Numerical Invariants of Totally Imaginary Quadratic Z[sqrt{p}]-orders, arXiv preprint arXiv:1603.02789, 2016

Crossrefs

Cf. A130230.