The least common multiple of consecutive arithmetic progression terms

TL;DR

This paper investigates the periodicity and smallest period of a generalized arithmetic function related to least common multiples of terms in arithmetic progressions, extending previous results from positive integers to general arithmetic progressions.

Contribution

It proves the periodicity of the function $g_{k,a,b}$ for all integers $a eq 0$, $b eq 0$, and determines its smallest period using $p$-adic analysis, extending prior work.

Findings

$g_{k,a,b}$ is periodic for all $a eq 0$, $b eq 0$

The smallest period of $g_{k,a,b}$ is explicitly determined

Extension of the Farhi-Kane theorem to general arithmetic progressions

Abstract

Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1,2,...,k)$ and conjectured that $\frac{{\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved this conjecture and determined the smallest period of $g_{k,1,0}$. For the general integers $a\ge 1$ and $b\ge 0$, it is natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic analysis of the periodic function $g_{k,a,b}$. Finally, we determine the smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from the set of positive integers to general arithmetic progressions.