On the equation 1k+2k + ••• + xk = yn for fixed x

JOURNAL OF NUMBER THEORY

卷:163

Article

Berczes, A.¹  Hajdu, L.¹  Miyazaki, T.²  Pink, I.^1,3 

[1] Univ Debrecen, Inst Math, POB 12, H-4010 Debrecen, Hungary

[2] Gunma Univ, Fac Sci & Engn, Gunma 3768515, Japan

[3] Salzburg Univ, Hellbrunnerstr 34-1, A-5020 Salzburg, Austria

关键词: Power sums;    Powers;    Schaffer's conjecture;   

DOI  :  10.1016/j.jnt.2015.11.008

来源: Elsevier

【 摘 要 】

We provide all solutions of the title equation in positive integers x, k, y, n with 1 <= x < 25 and n >= 3. For these values of the parameters, our result gives an affirmative answer to a related, classical conjecture of Schaffer. In our proofs we combine several tools: Baker's method (in particular, sharp bounds for the linear combinations of logarithms of two algebraic numbers), polynomial-exponential congruences and computational methods. (c) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free