On the Diophantine equation (x+1)(k) (x+2)(k) + . . . plus (lx)(k) = y(n)

SOYDAN G.

PUBLICATIONES MATHEMATICAE-DEBRECEN, vol.91, pp.369-382, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 91
  • Publication Date: 2017
  • Doi Number: 10.5486/pmd.2017.7679
  • Journal Name: PUBLICATIONES MATHEMATICAE-DEBRECEN
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.369-382
  • Keywords: Bernoulli polynomials, high degree equations
  • Bursa Uludag University Affiliated: Yes

Abstract

Let k, l >= 2 be fixed integers. In this paper, firstly, we prove that all solutions of the equation (x + 1)(k) + (x + 2)(k) + . . . + (lx)(k) = y(n) in integers x,y,n with x, y >= 1, n >= 2 satisfy n < C-1, where C-1 = C-1(l, k) is an effectively computable constant. Secondly, we prove that all solutions of this equation in integers x, y, n with x,y >= 1,n >= 2, k not equal 3 and I 0 (mod 2) satisfy max{x, y, n} < C-2, where C-2 is an effectively computable constant depending only on k and I.