THE LEAST COMMON MULTIPLE OF CONSECUTIVE ARITHMETIC PROGRESSION TERMS

Let k >= 0, a >= 1 and b >= 0 be integers. We define the arithmetic function g(k,a,b) for any positive integer n by g(k,a,b)(n) := (b + na)(b + (n + 1)a) ... (b + (n + k)a)/lcm(b + na, b + (n + 1)a, ..., b + (n + k)a). If we let a = 1 and b = 0, then g(k,a,b) becomes the arithmetic function that was previously introduced 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 lcm(1, 2, ..., k) and conjectured that (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 >= 1 and b >= 0, it is natural to ask the following interesting question: is g(k,a,b) periodic? If so, 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.