Amicable numbers and the bilinear diophantine equation

Abstract

2. Amicable Numbers. Method I. An amicable pair, (n1, n2) is defined by (4) o(n1) = o-((n2) = nli + n2 where adenotes the divisor sum function [1]. If we let (5) ni=Apq and n2= Br where p, q are primes relatively prime to some number A, and r is a prime relatively prime to a number B, (4) gives (6) S(p+1)(q+1) = T(r+1) =Apq+Br whereS = o-(A), T=o-(B). Solving for r and eliminating r gives (7) r = S(p + 1)(q + 1)/T -1 and (8) [AT-S(T-B)]pq-S(T-B)(p + q) = BT + S(T-B). Substituting in (2) we obtain (9) ad + bc = T[ABT + (A -B)(T-B)S]. Our procedure now is to find all factor pairs, M, N, of T[ABT + (A B) (T B) S] such that MN = T[ABT + (A B) (T -B) S]. For any such pair we solve the linear equations, (lOa) [AT-S(T-B)]p= N + S(T-B), (lOb) [AT-S(T-B)]q= M + S(T-B) given by (3) observing that (lOb) has a solution if and only if (lOa) has a solution