On the Diophantine equation (x+1)k + (x+2)k + ... + (2x)k = yn

JOURNAL OF NUMBER THEORY

卷:183

Article

Berczes, Attila¹  Pink, Istvan¹  Savas, Gamze²  Soydan, Gokhan² 

[1] Univ Debrecen, Inst Math, H-4032 Debrecen, Hungary

[2] Uludag Univ, Dept Math, TR-16059 Bursa, Turkey

关键词: Power sums;    Powers;    Polynomial-exponential congruences;    Linear forms in two logarithms;   

DOI  :  10.1016/j.jnt.2017.07.020

来源: Elsevier

【 摘 要 】

In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T-k(x) = (x + 1)(k) + (x + 2)(k) + ... + (2x)(k). Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. [6]), we show that for 2 <= x <= 13, k >= 1,y >= 2 and n >= 3 the title equation has no solutions. (C) 2017 Elsevier Inc. All rights reserved

【 授权许可】

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