OFFSET
0,5
COMMENTS
Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
(ii) Any positive integer n can be written as 4*x^4 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that 169*y^2 - 444*y*z + 396*z^2 (or 289*y^2 - 654*y*z + 401*z^2) is a square.
This is much stronger than Lagrange's four-square theorem, and we have verified parts (i) and (ii) of the conjecture for n up to 10^7 and 10^6 respectively.
By the linked JNT paper, any nonnegative integer n can be written as 4*x^4 + y^2 + z^2 + w^2 with x,y,z,w integers, and we can also write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and (y-z)*(y-2*z) = 0. Whether y = z or y = 2*z, the number 79*y^2 - 220*y*z + 205*z^2 is definitely a square.
See also A282933 for a similar conjecture.
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.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.
EXAMPLE
a(2) = 1 since 2 = 4*0^4 + 1^2 + 1^2 + 0^2 with 79*1^2 - 220*1*1 + 205*1^2 = 8^2.
a(35) = 1 since 35 = 4*0^4 + 3^2 + 1^2 + 5^2 with 79*3^2 - 220*3*1 + 205*1^2 = 16^2.
a(119) = 1 since 119 = 4*1^4 + 9^2 + 3^2 + 5^2 with 79*9^2 - 220*9*3 + 205*3^2 = 48^2.
a(124) = 1 since 124 = 4*1^4 + 4^2 + 2^2 + 10^2 with 79*4^2 - 220*4*2 + 205*2^2 = 18^2.
a(1564) = 1 since 1564 = 4*3^4 + 14^2 + 30^2 + 12^2 with 79*14^2 - 220*14*30 + 205*30^2 = 328^2.
a(4619) = 1 since 4619 = 4*2^4 + 51^2 + 27^2 + 35^2 with 79*51^2 - 220*51*27 + 205*27^2 = 228^2.
a(6127) = 1 since 6127 = 4*5^4 + 49^2 + 35^2 + 1^2 with 79*49^2 - 220*49*35 + 205*35^2 = 252^2.
a(7119) = 1 since 7119 = 4*1^4 + 51^2 + 17^2 + 65^2 with 79*51^2 - 220*51*17 + 205*17^2 = 272^2.
a(9087) = 1 since 9087 = 4*3^4 + 61^2 + 71^2 + 1^2 with 79*61^2 - 220*61*71 + 205*71^2 = 612^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-4x^4-y^2-z^2]&&SQ[79y^2-220*y*z+205z^2], r=r+1], {x, 0, (n/4)^(1/4)}, {y, 0, Sqrt[n-4x^4]}, {z, 0, Sqrt[n-4x^4-y^2]}]; Print[n, " ", r]; Continue, {n, 0, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 25 2017
STATUS
approved