A CONJECTURE CONCERNING THE RECIPROCAL PARTITION THEORY

Abstract. In this paper we prove that there exist infInitelv many disjoint sets of posItIve integers which the sum of whose reciprocals is equal to unity. Key words. disjoint set of positive integers, sum of reiprocals: unity. In [1] and [2] , Murthy proposed the following conjecture. Conjecture. There are infmitely many disjoint sets of positive integers \vhich the sum of \vhose reciprocals IS equal to unity. In this paper we completely verify the mentioned conjecture. For any pOSItIve integer n with n ~ 3: let A(n)= { a(n, 1) , a(n,2),... a(n,n)} be a disjoint set of positive integers having n elements, "where a(n,k) (k=1,2..., n) satisfy (1) a(3:1)=2, a(3,2)=3: a(3))=6