Category Archives: number theory

A Quasiperfect Number Must Be an Odd Perfect Square

Let be the sum of divisors function.  For example, , as the divisors of 5 are 1 and 5.  Similarly, , as the divisors of 6 are 1, 2, 3, and 6. A perfect number is one that is equal … Continue reading

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The Validity of Mathematical Induction

Suppose you have some statement .  Mathematical induction says that the following is sufficient to prove that is true for all natural numbers k. is true. For any natural number k, if is true, then is true. The idea is that the first … Continue reading

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Generalized Binomial Coefficients from Multiplicative and Divisible Functions

Given a function f from the natural numbers to the natural numbers, one way to generalize the binomial coefficient is via . The usual binomial coefficient of course has f as the identity function . Question: What kinds of functions f guarantee that … Continue reading

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Integrality of the Catalan Numbers via Kummer’s Theorem

Why is the nth Catalan number, , an integer?  If you know one of its combinatorial interpretations, then the answer is clear, but how do you get integrality strictly from this formula?  In this post I’m going to discuss how one can … Continue reading

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A Combinatorial Proof for the Power Sum

Probably the most common formula for the power sum is the one involving binomial coefficients and Bernoulli numbers , sometimes called Faulhaber’s formula: Historically, this, or a variant of it, was the first general formula for the power sum. There … Continue reading

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