A271518 - OEIS

%I #65 Mar 31 2021 22:34:58

%S 1,2,2,2,2,1,1,1,1,3,3,2,2,2,4,2,2,5,5,3,2,2,2,3,1,5,5,2,2,5,8,1,2,6,

%T 3,3,2,3,7,5,2,8,6,1,4,6,6,2,2,6,9,5,4,3,7,6,2,6,7,5,2,1,6,6,2,10,9,6,

%U 3,3,6,2,3,8,12,5,5,7,11,5,1

%N Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x + 3*y + 5*z a square, where w, x, y and z are nonnegative integers.

%C Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4^k*6 (k = 0,1,2,...), 16^k*m (k = 0,1,2,... and m = 5, 7, 8, 31, 43, 61, 116).

%C (ii) Any integer n > 15 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 6*x + 10*y + 12*z a square.

%C (iii) Each nonnegative integer n not among 7, 15, 23, 71, 97 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 2*x + 6*y + 10*z a square. Also, any nonnegative integer n not among 7, 43, 79 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 3*x + 5*y + 6*z a square.

%C See also A271510 and A271513 for related conjectures.

%C a(n) > 0 verified for all n <= 3*10^7. - _Zhi-Wei Sun_, Nov 28 2016

%C Qing-Hu Hou at Tianjin Univ. has verified a(n) > 0 and parts (ii) and (iii) of the above conjecture for n up to 10^9. - _Zhi-Wei Sun_, Dec 04 2016

%C The conjecture that a(n) > 0 for all n = 0,1,2,... is called the 1-3-5-Conjecture and the author has announced a prize of 1350 US dollars for its solution. - _Zhi-Wei Sun_, Jan 17 2017

%C Qing-Hu Hou has finished his verification of a(n) > 0 for n up to 10^10. - _Zhi-Wei Sun_, Feb 17 2017

%C The 1-3-5 conjecture was finally proved by António Machiavelo and Nikolaos Tsopanidis in a JNT paper published in 2021. This is a great achivement! - _Zhi-Wei Sun_, Mar 31 2021

%H Zhi-Wei Sun, <a href="/A271518/b271518.txt">Table of n, a(n) for n = 0..10000</a>

%H António Machiavelo and Nikolaos Tsopanidis, <a href="https://arxiv.org/abs/2003.02592">Zhi-Wei Sun's 1-3-5 Conjecture and Variations</a>, arXiv:2003.02592 [math.NT], 2020.

%H António Machiavelo and Nikolaos Tsopanidis, <a href="https://doi.org/10.1016/j.jnt.2020.10.001">Zhi-Wei Sun's 1-3-5 Conjecture and Variations</a>, J. Number Theory 222 (2021), 1-20.

%H António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, <a href="https://arxiv.org/abs/2005.13526">Report on Zhi-Wei Sun's "1-3-5 conjecture" and some of its refinements</a>, arXiv:2005.13526 [math.NT], 2020.

%H António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, <a href="https://doi.org/10.1016/j.jnt.2021.01.001">Report on Zhi-Wei Sun's 1-3-5 conjecture and some of its refinements</a>, J. Number Theory 222 (2021), 21-29.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.NT], 2016.

%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 Conjecture 4.3(i) and Remark 4.3.)

%e a(5) = 1 since 5 = 2^2 + 1^2 + 0^2 + 0^2 with 1 + 3*0 + 5*0 = 1^2.

%e a(6) = 1 since 6 = 2^2 + 1^2 + 1^2 + 0^2 with 1 + 3*1 + 5*0 = 2^2.

%e a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 + 3*1 + 5*1 = 3^2.

%e a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.

%e a(24) = 1 since 24 = 4^2 + 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.

%e a(31) = 1 since 31 = 1^2 + 5^2 + 2^2 + 1^2 with 5 + 3*2 + 5*1 = 4^2.

%e a(43) = 1 since 43 = 1^2 + 1^2 + 5^2 + 4^2 with 1 + 3*5 + 5*4 = 6^2.

%e a(61) = 1 since 61 = 6^2 + 0^2 + 0^2 + 5^2 with 0 + 3*0 + 5*5 = 5^2.

%e a(116) = 1 since 116 = 10^2 + 4^2 + 0^2 + 0^2 with 4 + 3*0 + 5*0 = 2^2.

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

%t Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+3y+5z],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,0,80}]

%Y Cf. A000118, A000290, A270969, A271510, A271513, A273294, A273302, A276533, A278560.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Apr 09 2016