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

被引：5

作者：

Berczes, A. ^{[1
]}

Hajdu, L. ^{[1
]}

Miyazaki, T. ^{[2
]}

Pink, I. ^{[1
,3
]}

机构：

[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

来源：

JOURNAL OF NUMBER THEORY | 2016年 / 163卷

基金：

奥地利科学基金会;

关键词：

Power sums; Powers; Schaffer's conjecture; VALUES;

D O I：

10.1016/j.jnt.2015.11.008

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

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.

引用

页码：43 / 60

页数：18

共 17 条

[1] On the Diophantine equation 1k+2k +...+ xk = yn [J].

Bennett, MA ;

Gyory, K ;

Pintér, A .

COMPOSITIO MATHEMATICA, 2004, 140 (06) :1417-1431

[2] The Magma algebra system .1. The user language [J].

Bosma, W ;

Cannon, J ;

Playoust, C .

JOURNAL OF SYMBOLIC COMPUTATION, 1997, 24 (3-4) :235-265

[3] Equal values of standard counting polynomials [J].

Gyoery, Kalman ;

Kovacs, Tunde ;

Peter, Gyoengyver ;

Pinter, Akos .

PUBLICATIONES MATHEMATICAE-DEBRECEN, 2014, 84 (1-2) :259-277

[4]

Györy K, 2003, PUBL MATH-DEBRECEN, V62, P403

[5]

Gyory K., 1980, ACTA ARITH, V37, P234

[6] On a conjecture of Schaffer concerning the equation 1k + ... + xk = yn [J].

Hajdu, L. .

JOURNAL OF NUMBER THEORY, 2015, 155 :129-138

[7] A computational approach for solving y2=1k+2k+•••+xk [J].

Jacobson, MJ ;

Pintér, A ;

Walsh, PG .

MATHEMATICS OF COMPUTATION, 2003, 72 (244) :2099-2110

[8] Linear forms in two logarithms and interpolation determinants II [J].

Laurent, Michel .

ACTA ARITHMETICA, 2008, 133 (04) :325-348

[9]

Lucas E., 1875, NOUVELLES ANN MATH, V14, P336

[10]

Lucas E., 1877, NOUVELLES ANN MATH, V16, P429

← 1 2 →