A338272 Number of ways to write 4*n + 1 as x^2 + y^2 + z^2 + w^2 (0 <= x <= y and 0 <= z <= w) such that x*y + 32*z*w is a square.

2, 3, 4, 2, 4, 4, 4, 3, 3, 3, 4, 5, 4, 3, 3, 3, 5, 4, 4, 2, 8, 6, 6, 2, 4, 6, 4, 4, 5, 4, 5, 6, 5, 1, 3, 4, 6, 6, 6, 4, 6, 3, 6, 2, 3, 4, 5, 8, 4, 3, 6, 5, 5, 3, 2, 4, 8, 4, 3, 3, 5, 6, 4, 2, 5, 10, 6, 5, 4, 3, 6, 4, 7, 5, 8, 2, 6, 6, 4, 3, 7

Offset: 0

Author

Zhi-Wei Sun, Oct 19 2020

Keywords

• nonn

Comments

Conjecture: Any positive integer congruent to 1 modulo 4 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that a*x*y + b*z*w is a square, provided that (a,b) is among the following ordered pairs: (1,-2), (1,2), (1,10), (1,32), (2,-1), (2,10), (2,14), (2,16), (2,36), (3,4), (4,-2), (4,18), (6,18), (8,9), (9,10), (16,-2).

a(n) > 0 for all n = 0..10^5.

Examples

			a(33) = 1, and 4*33 + 1 = 4^2 + 9^2 + 0^2 + 6^2 with 4*9 + 32*0*6 = 6^2.
a(364) = 1, and 4*364 + 1 = 16^2 + 25^2 + 0^2 + 24^2 with 16*25 + 32*0*24 = 20^2.
a(1319) = 1, and 4*1319 + 1 = 20^2 + 36^2 + 10^2 + 59^2 with 20*36 + 32*10*59 = 140^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].
• Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893. See also arXiv:1701.05868 [math.NT].
• Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.

Crossrefs

Cf. A000118, A000290, A270073.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4n+1-x^2-y^2-z^2]&&SQ[x*y+32*z*Sqrt[4n+1-x^2-y^2-z^2]],r=r+1],
{x,0,Sqrt[2n]},{y,x,Sqrt[4n+1-x^2]},{z,0,Sqrt[(4n+1-x^2-y^2)/2]}];tab=Append[tab,r],{n,0,80}];tab