A350714 Least positive integer m such that m^12*n = x^4 + y^3 + z^2 for some nonnegative integers x,y,z.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset: 0

Author

Zhi-Wei Sun, Feb 02 2022

Keywords

• nonn

Comments

4-3-2 Conjecture: a(n) exists for any nonnegative integer n. Equivalently, each nonnegative rational number can be written x^4 + y^3 + z^2 with x,y,z nonnegative rational numbers.

Note that m/n = (m*n^11)/n^12 for any positive integers m and n.

a(n) <= 4 for n <= 40000 with the only exception a(23710) = 5.

a(n) <= 4 for n = 77000..100000, and a(n) = 4 for n = 78367, 79479, 83494, 84694, 85979, 86822, 87395, 87814, 90047, 90278, 92891, 93715.

Qing-Hu Hou verified a(n) <= 4 for 40000 < n < 77000. - Zhi-Wei Sun Feb 04 2022

a(n) <= 5 for 10^5 < n <= 2*10^5, and a(n)=5 for n=107206, 117615, and 148079. - Qing-Hu Hou, Feb 05 2022

Examples

			a(6) = 1 with 1^12*6 = 1^4 + 1^3 + 2^2.
a(7) = 2 with 2^12*7 = 2^4 + 15^3 + 159^2.
a(75) = 4 with 4^12*75 = 122^4 + 1007^3 + 3951^2.
a(1140) = 3 with 3^12*1140 = 0^4 + 531^3 + 21357^2.
a(23710) = 5 with 5^12*23710 = 217^4 + 17897^3 + 232166^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
• Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Crossrefs

Cf. A000290, A000578, A000583, A280356.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[m=1;Label[bb];k=m^12;Do[If[SQ[k*n-x^4-y^3],tab=Append[tab,m];Goto[aa]],{x,0,(k*n)^(1/4)},{y,0,(k*n-x^4)^(1/3)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]