OFFSET
0,1
COMMENTS
Conjecture: Any integer m can be written as x^4 - y^3 + z^2, where x, y and z are positive integers.
This is slightly stronger than the conjecture in A266003.
If n is a positive square, then a(n) = 1. - Altug Alkan, Dec 23 2015
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.
EXAMPLE
a(0) = 8 since 0 = 4^4 - 8^3 + 16^2.
a(6) = 139 since 6 = 36^4 - 139^3 + 1003^2.
a(15) = 1411 since 15 = 119^4 - 1411^3 + 51075^2.
a(11019) = 71383 since 11019 = 4325^4 - 71383^3 + 3719409^2.
MATHEMATICA
SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
Do[y=1; Label[bb]; Do[If[SQ[n+y^3-x^4], Print[n, " ", y]; Goto[aa]], {x, 1, (n+y^3)^(1/4)}]; y=y+1; Goto[bb]; Label[aa]; Continue, {n, 0, 70}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 22 2015
STATUS
approved