A note on Diophantine systems involving three symmetric polynomials

JOURNAL OF NUMBER THEORY

卷:133

Article

Ulas, Maciej

关键词: Symmetric polynomials;    Elliptic curves;   

DOI  :  10.1016/j.jnt.2013.04.009

来源: Elsevier

【 摘 要 】

Let (X) over bar (n) = (x(1), ..., x(n)) and sigma(i)((X) over bar (n)) = Sigma x(k1) ... x(k1) be i-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers a, b, c such that for each 1 <= i <= n the system of Diophantine equations sigma(i)((X) over bar (2n)) = a, sigma(2n-1) ((X) over bar (2n)) = b, sigma(2n)((X) over bar (2n)) = c has infinitely many rational solutions. This result extends the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. In particular, we prove that for each k there are at least k n-tuples of integers with the same sum of i-th powers for i = 1,2,3. Similar result is proved for i = 1,2,4 and i = 1,1,2. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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