OFFSET
0,4
COMMENTS
6-6-3-3 Conjecture: Each rational number can be written as u^6 - v^6 + x^3 - y^3 with u,v,x,y nonnegative rational numbers. Moreover, a(n) exists for any nonnegative integer n.
As a/b = (a*b^5)/b^6 for any integer a and nonzero integer b, the second assertion in the conjecture implies the first one.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..1000
EXAMPLE
a(3) = 21 with 21^6*3 = 22^6 + 956^3 - (30^6 + 93^3) and 30^6 + 93^3 <= 21^6*3^2.
a(67) = 15 with 15^6*67 = 21^6 + 1091^3 - (15^6 + 848^3) and 15^6 + 848^3 <= 15^6*67^2.
a(564) = 14 with 14^6*564 = 69^6 + 4415^3 - (16^6 + 5746^3) and 16^6 + 5746^3 <= 14^6*564^2.
a(949) = 18 with 18^6*949 = 7^6 + 11784^3 - (11^6 + 11706^3) and 11^6 + 11706^3 <= 18^6*949^2.
MATHEMATICA
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={}; Do[m=1; Label[bb]; k=m^6; Do[If[CQ[k*n+x^6+y^3-z^6], tab=Append[tab, m]; Goto[aa]],
{x, 0, m*n^(1/3)}, {y, 0, (k*n^2-x^6)^(1/3)}, {z, 0, (k*n+x^6+y^3)^(1/6)}]; m=m+1; Goto[bb]; Label[aa], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 07 2022
STATUS
approved