A Note on the Factorization of Integers

Abstract

The purpose of this paper is to demonstrate the usefulness of the binary quadratic form, (1) x2 + pqy2 in integers x and y with integral coefficients, in the factorization of odd integers of the form 4k + 1, where k is a positive integer. We begin by introducing the following two lemmas. Lemma 1. If Al is a positive integer of the form 4k + 1 and r1 and r2 are any pair of positive integral roots of M where M = r1r2, there existintegers u, m, v, n, p, and q such that: (i) r=pu 2+qv2 (ii) r2= pm2 + qn2 Proof: Let p = v = n = 1