A356488 Numbers k such that the equation x^2 - k*y^4 = -1 has a solution for which |y| > 2.

2, 53, 314, 1042, 1685, 1825, 3281, 4586, 5521, 6770, 8597, 9050, 11509, 13858, 17498, 20369, 24737, 28085, 28130, 29041, 31226, 33226, 37141, 37585, 42965, 47402, 49205, 53954, 57125, 58913, 66193, 71674, 79682, 85685, 94421, 100946, 110410, 113290, 115202

Offset

1,1

Comments

For k > 2, the equation x^2 - k*y^4 = -1 has at most one positive integer solution. If this solution (x, y) exists, we have v = y^2, where v is the smallest integer satisfying the Pell equation u^2 - k*v^2 = -1 (A130227).

Links

Table of n, a(n) for n=1..39.

Chen Jian Hua and Paul Voutier, Complete solution of the diophantine equation X^2 + 1 = dY^4 and a related family of quartic Thue equations, arXiv:1401.5450 [math.NT], 2014-2018.

Example

The equation x^2 - 2*y^4 = -1 has only two positive solutions (1, 1) and (239, 13), so 2 is in the sequence.

Crossrefs

Cf. A031396, A130227, A182468.

Sequence in context: A281227 A142477 A119112 * A109791 A265441 A354406

Adjacent sequences: A356485 A356486 A356487 * A356489 A356490 A356491

Keyword

nonn

Author

Jinyuan Wang, Aug 09 2022

Status

approved