OFFSET
1,1
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.
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann and Hugo Pfoertner, Dec 09 2021
STATUS
approved