The numbers that can be represented by a special cubic polynomial

Doo Sung Park; Seung Jin Bang; Jung Oh Choi

doi:10.4134/CKMS.2010.25.2.167

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2010; 25(2): 167-171

Printed June 1, 2010

https://doi.org/10.4134/CKMS.2010.25.2.167

The numbers that can be represented by a special cubic polynomial

Doo Sung Park, Seung Jin Bang, and Jung Oh Choi

California Institute of Technology, Ajou University, and Saetbyeol Middle School

Abstract

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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