Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

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In This paper there are given thirteen proofs that √2 is irrational. Indications are given as to whether the method employed extends to proving the irrationality of other square roots or of roots of higher order. In addition, a reference is provided to an incorrect proof recently published and its criticism.

Arithmetic sums of the formwhere f is an arithmetic function and [ ] is the greatest integer function frequently occur in various situations in the theory of numbers and have much of interest in their own right. Two instances appear in the well-known results.