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abundant integers

Source: Ireland 1997

July 3, 2009
number theory unsolvednumber theory

Problem Statement

Given a positive integer n n, denote by σ(n) \sigma (n) the sum of all positive divisors of n n. We say that n n is abundant abundant if σ(n)>2n. \sigma (n)>2n. (For example, 12 12 is abundant since \sigma (12)\equal{}28>2 \cdot 12.) Let a,b a,b be positive integers and suppose that a a is abundant. Prove that ab ab is abundant.
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