THE NUMBERS THAT CAN BE REPRESENTED BY A SPECIAL CUBIC POLYNOMIAL

Title & Authors
THE NUMBERS THAT CAN BE REPRESENTED BY A SPECIAL CUBIC POLYNOMIAL
Park, Doo-Sung; Bang, Seung-Jin; Choi, Jung-Oh;

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 is a cubic residue (mod p), $\small{3\;{\nmid}\;m}$, p || n if and only if there are integers x, y, z such that $\small{x^3\;+\;dy^3\;+\;d^2z^3\;-\;3dxyz\;=\;n}$.
Keywords
number theory;
Language
English
Cited by
References
1.
T. Hungerford, Algebra, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.- London, 1974.

2.
F. Lemmermeyer, Reciprocity Laws, Springer-Verlag, Berlin, 2000.

3.
D. A. Marcus, Number Fields, Springer-Verlag, New York-Heidelberg, 1977.

4.
I. Niven, H. S. Zuckerman, and H. Montgomery, An Introduction to the Theory of Numbers, John Wiley & Sons, Inc., New York, 1991.