A349663 Positive numbers x for which x^2 can be expressed as z^2 - y^4 with y != 0.

3, 12, 15, 17, 27, 30, 40, 42, 48, 60, 63, 68, 75, 77, 90, 95, 99, 105, 108, 112, 120, 130, 135, 140, 147, 153, 156, 160, 165, 168, 192, 195, 220, 240, 243, 252, 270, 272, 273, 300, 301, 308, 312, 315, 323, 350, 360, 363, 375, 378, 380, 396, 399, 420, 425, 432, 448, 462, 480, 495, 507

Offset: 1

Author

Karl-Heinz Hofmann and Hugo Pfoertner, Dec 09 2021

Keywords

• nonn

Comments

This sequence is closely related to A271576.

Conditions to be satisfied for a solution:

- z cannot be a square.

- z must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).

- If z has prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.

- If z is even, x and y must be even too.

- The lower limit of the ratio x/y is sqrt(2).

Multiple solutions are possible; e.g., term 420 has 5 solutions.

Examples

			The 5 solutions corresponding to a(54) = 420 are 420^2 = 176400 = 444^2 - 12^4 = 580^2 - 20^4 = 609^2 - 21^4 = 1295^2 - 35^4 = 3164^2 - 56^4.
		

Crossrefs

Cf. A000290, A000583, A271576, A002144, A002145.