Power values of sums of certain products of consecutive integers and\n related results

2018/09/12 by Szabolcs Tengely, Tengely, Szabolcs, Maciej Ulas +1
Mathematics · #11D41 #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1809.04304

openalex publication_date 2018/09/12 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28

Abstract

Let n be a non-negative integer and put p_n(x)=\∏_i=0ⁿ(x+i). In\nthe first part of the paper, for given n, we study the existence of integer\nsolutions of the Diophantine equation \ny^m=p_n(x)+
sum_i=1^kp_a_i(x), where m\∈ N_{\≥ 2} and\na₁<a₂<\… <a_k<n. This equation can be considered as a\ngeneralization of the Erd Hos-Selfridge Diophantine equation y^m=p_n(x).\nWe present some general finiteness results concerning the integer solutions of\nthe above equation. In particular, if n\≥ 2 with a₁\≥ 2, then our\nequation has only finitely many solutions in integers. In the second part of\nthe paper we study the equation y^m=
sum_i=1^kp_a_i(x_i), for\nm=2, 3, which can be seen as an additive version of the equation considered\nby Erd Hos and Graham. In particular, we prove that if m=2, a₁=1 or\nm=3, a₂=2, then for each k-1 tuple of positive integers (a₂,\…,\na_k) there are infinitely many solutions in integers.\n

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