A343917 Positive integers m with 2*m^2 - 2^4 = x^4 + y^4 for some nonnegative integers x and y with |x-y| > 2.

284, 1388, 2139, 4772, 8556, 8971, 10836, 21163, 28847, 45707, 54507, 71292, 73348, 95127, 101503, 104228, 131388, 136148, 263172, 350076, 638164, 982292, 1532148, 1687828, 1705407, 1958924, 2082188, 2299364, 2360347, 2728379, 3202356, 4042799, 5046771, 5165332, 5235323, 5560627, 7191079, 7740547, 8041364

Offset: 1

Author

Zhi-Wei Sun, May 04 2021

Keywords

• nonn

Comments

Conjecture: The sequence has infinitely many terms.

It is easy to see that no term is divisible by 5. In the b-file we list all the 62 terms not exceeding 10^8.

Note that 2*(n^2+3)^2 - 2^4 = (n+1)^4 + (n-1)^4 with (n+1) - (n-1) = 2. This implies that any integer n > 4 can be written as x + y + 2^(z-1) with x,y,z positive integers such that x^4 + y^4 + (2^z)^4 is twice a square.

See also A343913 for a similar conjecture.

Examples

			a(1) = 284, and 2*284^2 - 2^4 = 20^4 + 6^4 with |20-6| > 2.
a(62) = 97077407, and 2*97077407^2 - 2^4 = 18848045899687282 = 11563^4 + 5583^4 with |11563-5583| > 2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..62

Crossrefs

Cf. A000290, A000583, A003336, A230121, A230747, A340274, A343862, A343897, A343913.

Programs

Mathematica

QQ[n_]:=IntegerQ[n^(1/4)];
n=0;Do[Do[If[QQ[2*m^2-16-x^4]&&(2*m^2-16-x^4)^(1/4)-x>2,n=n+1;Print[n," ",m];Goto[aa]],{x,0,(m^2-8)^(1/4)}];Label[aa],{m,3,8041364}]