On the least common multiple of shifted powers

2021/03/14 by Carlo Sanna, Sanna, Carlo
Computer Science · Mathematics · #11B39 (Primary) 11B37 #11N37 (Secondary) #Analytic Number Theory Research #Coding theory and cryptography #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2103.07967

openalex publication_date 2021/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a ≥ 2 be an integer. We prove that for every periodic sequence (s_n)_{n ≥ 1} in \-1, +1\ there exists an effectively computable rational number C_s > 0 such that \loglcm(a + s₁, a² + s₂, …, aⁿ + s_n) ∼ C_s ⋅ (log a)/(π²) ⋅ n² , as n → +∞, where lcm denotes the least common multiple. Furthermore, we show that if (s_n)_{n ≥ 1} is a sequence of independent and uniformly distributed random variables in \-1, +1\ then \loglcm(a + s₁, a² + s₂, …, aⁿ + s_n) ∼ 6 Li₂ (\tfrac12) ⋅ (log a)/(π²) ⋅ n² , with probability 1 - o(1), as n → +∞, where Li₂ is the dilogarithm function.

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