The numbers that can be represented by a special cubic polynomial
Commun. Korean Math. Soc. 2010 Vol. 25, No. 2, 167-171
Published online June 1, 2010
Doo Sung Park, Seung Jin Bang, and Jung Oh Choi
California Institute of Technology, Ajou University, and Saetbyeol Middle School
Abstract : We will show that if $d$ is a cubefree integer and $n$ is an integer, then with some suitable conditions, there are no primes $p$ and a positive integer $m$ such that $$ d \text{ is a cubic residue} \pmod p, ~3\nmid m,~p\parallel n$$ if and only if there are integers $x,y,z$ such that \[x^3+dy^3+d^2z^3-3dxyz=n.\]
Keywords : number theory
MSC numbers : 11D25
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