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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. 1

%I #9 Oct 20 2020 08:20:34

%S 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,

%T 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,

%U 4,3,6,4,7,5,8,2,6,6,4,3,7

%N 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.

%C 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).

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

%H Zhi-Wei Sun, <a href="/A338272/b338272.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190. See also <a href="http://arxiv.org/abs/1604.06723">arXiv:1604.06723 [math.NT]</a>.

%H Zhi-Wei Sun, <a href="https://doi.org/10.1142/S1793042119501045">Restricted sums of four squares</a>, Int. J. Number Theory 15(2019), 1863-1893. See also <a href="http://arxiv.org/abs/1701.05868">arXiv:1701.05868 [math.NT]</a>.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020.

%e a(33) = 1, and 4*33 + 1 = 4^2 + 9^2 + 0^2 + 6^2 with 4*9 + 32*0*6 = 6^2.

%e a(364) = 1, and 4*364 + 1 = 16^2 + 25^2 + 0^2 + 24^2 with 16*25 + 32*0*24 = 20^2.

%e a(1319) = 1, and 4*1319 + 1 = 20^2 + 36^2 + 10^2 + 59^2 with 20*36 + 32*10*59 = 140^2.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t 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],

%t {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

%Y Cf. A000118, A000290, A270073.

%K nonn

%O 0,1

%A _Zhi-Wei Sun_, Oct 19 2020

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)