Some Number-Theoretic Products

摘要

For a positive integer n, let P(n) and P′ (n) be the products of all elements in the finite sets {x: 1 ≤ x ≤ n, (x, n) = 1} and {x : 1 ≤ x ≤ n/2, (x, n) = 1}, respectively. In this article, we verify the formula for P(n) and use it to establish the formula for P′ (n). Explicit formulae for both P (pa) and P′ (pa), where pa is a prime power, are also derived.