Sums of products of congruence classes and of arithmetic progressions
Abstract
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+ m) for some i,j,k,∈ 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).
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