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A350714
Least positive integer m such that m^12*n = x^4 + y^3 + z^2 for some nonnegative integers x,y,z.
5
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
0,8
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
LINKS
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.
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 02 2022
STATUS
approved