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indeterminate equation

Source: China south east mathematical olympiad 2006 day2 problem 7

July 4, 2013
number theory unsolvednumber theory

Problem Statement

(1) Find the number of positive integer solutions (m,n,r)(m,n,r) of the indeterminate equation mn+nr+mr=2(m+n+r)mn+nr+mr=2(m+n+r). (2) Given an integer k(k>1)k (k>1), prove that indeterminate equation mn+nr+mr=k(m+n+r)mn+nr+mr=k(m+n+r) has at least 3k+13k+1 positive integer solutions (m,n,r)(m,n,r).
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