OFFSET

0,1

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.

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.

EXAMPLE

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.

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

CROSSREFS

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 19 2020

STATUS

approved