On the Diophantine equation (&ITx&IT+1)&ITk&IT + (&ITx&IT+2)&ITk&IT + ... + (2&ITx&IT)&ITk&IT = &ITy(n)&IT


Berczes A., Pink I., Savas G., SOYDAN G.

JOURNAL OF NUMBER THEORY, vol.183, pp.326-351, 2018 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 183
  • Publication Date: 2018
  • Doi Number: 10.1016/j.jnt.2017.07.020
  • Journal Name: JOURNAL OF NUMBER THEORY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.326-351
  • Keywords: Power sums, Powers, Polynomial-exponential congruences, Linear forms in two logarithms, SUMS
  • Bursa Uludag University Affiliated: Yes

Abstract

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