Integer partitions into diophantine pairs

Abstract

Let n, a and b be positive integers. The pair (a; b) is called an integer partition of n into Diophantine pair if n = a+b, ab+1 is a perfect square and a > b. In this paper we give, for any positive integer n, a closed formula for the number of integer partitions into Diophantine pairs.

Mathematics Subject Classication (2010): Primary 05A17; Secondary 11P83.

Key words: Integer partitions, Diophantine pairs, τ (n), qD(n;m), qD(n; 2):