A certain reciprocal power sum is never an integer

2018/12/20 by Junyong Zhao, Zhao, Junyong, Shaofang Hong +3
Mathematics · #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1812.08705

openalex publication_date 2018/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By (ℤ⁺)^∞ we denote the set of all the infinite sequences S=\s_i\_i=1^∞ of positive integers (note that all the s_i are not necessarily distinct and not necessarily monotonic). Let f(x) be a polynomial of nonnegative integer coefficients. Let S_n:=\s₁, ..., s_n\ and H_f(S_n):=∑_k=1ⁿ\frac1f(k)^s_k. When f(x) is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence S of positive integers, H_f(S_n) is never an integer if n≥ 2. Now let degf(x)≥ 2. Clearly, 00, there are positive integers n₁ and n₂ and infinite sequences S⁽¹⁾ and S⁽²⁾ of positive integers such that 1-ε

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