A351306 Least positive integer m such that m^4*n = u^4 + v^4 - (x^4 + y^4) for some nonnegative integers u,v,x,y with x^4 + y^4 <= m^4*n^2.

1, 1, 1, 10, 2, 2, 2, 4, 6, 4, 2, 2, 4, 8, 1, 1, 1, 1, 2, 2, 2, 2, 10, 2, 2, 2, 2, 10, 10, 2, 1, 1, 1, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 10, 2, 2, 1, 1, 5, 1, 1, 4, 10, 10, 2, 2, 6, 10, 4, 4, 2, 4, 1, 3, 1, 1, 1, 10, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 4, 10, 2, 2, 4, 6, 6, 1, 1, 1, 1, 5, 2, 2

Offset: 0

Author

Zhi-Wei Sun, Feb 06 2022

Keywords

• nonn

Comments

Conjecture: Each n >= 0 can be written as u^4 + v^4 - (x^4 + y^4), where u,v,x,y are rational numbers with x^4 + y^4 <= n^2. In other words, a(n) exists for any nonnegative integer n.

A known result of R. Norrie states that any rational number can be written as u^4 + v^4 - (x^4 + y^4) with u,v,x,y rational numbers.

Examples

			a(3) = 10 with 10^4*3 = 8^4 + 13^4 - (4^4 + 7^4) and 4^4 + 7^4 <= 10^4*3^2.
a(242) = 15 with 15^4*242 = 73^4 + 153^4 - (36^4 + 154^4) and 36^4 + 154^4 <= 15^4*242^2.
a(248) = 28 with 28^4*248 = 95^4 + 270^4 - (52^4 + 269^4) and 52^4 + 269^4 <= 28^4*248^2.
a(313) = 30 with 30^4*313 = 37^4 + 128^4 - (7^4 + 64^4) and 7^4 + 64^4 <= 30^4*313^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..800
• Tito Piezas III, A Collection of Algebraic Identities, 001b: Assorted identities, Part 2 (Waring-like problems).
• Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Crossrefs

Cf. A000583, A004831, A351312.

Programs

Mathematica

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
tab={};Do[m=1;Label[bb];k=m^4;Do[If[QQ[k*n+x^4+y^4-z^4],tab=Append[tab,m];Goto[aa]],
{x,0,m*(n^2/2)^(1/4)},{y,x,(k*n^2-x^4)^(1/4)},{z,0,((k*n+x^4+y^4)/2)^(1/4)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]