Sums of Products of Congruence Classes and of Arithmetic Progressions

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a+im : i ∈ N0}. For positive integers a, b, c, d,m the sum of products set Rm(a)Rm(b)+Rm(c)Rm(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+lm) for some i, j, k, l ∈ Z}. It is proved that if gcd(a, b, c, d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab + cd), and that the sum of products set Pm(a)Pm(b) + Pm(c)Pm(d) eventually coincides with the infinite arithmetic progression Pm(ab + cd). 1. Sums of product sets Let Z denote the set of integers and N0 the set of nonnegative integers. For every prime p and integer n, we denote by ordp(n) the greatest integer k such that p divides n. Let X and Y be sets of integers. These sets eventually coincide, denoted X ∼ Y, if there is an integer n0 such that, for all n ≥ n0, we have n ∈ X if and only if n ∈ Y. We define the sumset X + Y = {x + y : x ∈ X, y ∈ Y }, the product set XY = {xy : x ∈ X, y ∈ Y }, and, for any integer δ, the dilation δ∗X = {δx : x ∈ X}. Let a, b, and m be integers with m ≥ 1. We denote the congruence class of a modulo m by Rm(a) = {a+ im : i ∈ Z}. For all a and b, we have Rm(a)Rm(b) ⊆ Rm(ab). This inclusion can be strict. For example, 53 ∈ R19(15) but 53 / ∈ R19(3)R19(5) since 53 is prime. Thus, the product of two congruence classes modulo m is not necessarily a congruence class modulo m. The case of sums of products of congruence classes is different. For all integers a, b, c, d, and m with m ≥ 1 we have Rm(a)Rm(b) +Rm(c)Rm(d) ⊆ Rm(ab+ cd). We shall prove that if gcd(a, b, c, d,m) = 1, then Rm(a)Rm(b) +Rm(c)Rm(d) = Rm(ab+ cd) Date: February 1, 2008. 2000 Mathematics Subject Classification. Primary 11A07, 11B25, 11B75.