A060829 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both squares; list all such pairs (x,y) with gcd(x,y) = 1 ordered by values of y; sequence gives x values.

5, 17, 13, 37, 65, 29, 101, 25, 145, 53, 197, 257, 85, 325, 41, 401, 125, 485, 73, 577, 173, 677, 65, 785, 61, 109, 229, 901, 1025, 293, 1157, 97, 1297, 365, 1445, 89, 1601, 85, 205, 445, 1765, 137, 1937, 533, 2117, 265, 2305, 629, 2501, 185, 2705, 733, 2917

Offset

0,1

References

Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

Links

Table of n, a(n) for n=0..52.

Formula

The solutions are given by x = r^2+2*r*k+2*k^2, y = 2*k*(k+r) with r >= 1, k >= 1, r odd, gcd(r, k) = 1.

Example

Pairs are [5, 4], [17, 8], [13, 12], [37, 12], [65, 16], [29, 20], [101, 20], ... E.g., 5-4=1^2, 5+4=3^2.
a(41) = 1765 because A120427(41) = 84 and we have gcd(1765,84)=1 and 1765-84 = 41^2 and 1765+84 = 43^2. - Sean A. Irvine, Jan 01 2023

Crossrefs

Cf. A060830, A061408, A061409, A120427.

Sequence in context: A302159 A070848 A213738 * A096896 A056850 A302631

Adjacent sequences: A060826 A060827 A060828 * A060830 A060831 A060832

Keyword

nonn

Author

N. J. A. Sloane, May 02 2001

Extensions

a(41) onward corrected by Sean A. Irvine, Jan 01 2023

Status

approved