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Category Archives: number theory
A Lesson on Converting Between Different Bases
We’re in the time of COVID19, and that has meant taking far more direct responsibility for my children’s learning than I ever have before. It’s been a lot of work, but it’s also been fun. In fact, I’ve been surprised … Continue reading
Posted in number theory
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Strong Induction Wasn’t Needed After All
Lately when I’ve taught the second principle of mathematical induction – also called “strong induction” – I’ve used the following example to illustrate why we need it. Prove that you can make any amount of postage of 12 cents or … Continue reading
Posted in number theory, proof techniques
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Arguments for 0.9999… Being Equal to 1
Recently I tried to explain to my 11yearold son why 0.9999… equals 1. The standard arguments for (at least the ones I’ve seen) assume more math background than he has. So I tried another couple of arguments, and they seemed … Continue reading
Posted in arithmetic, number theory
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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
Posted in number theory
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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
Posted in number theory, proof techniques
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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
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
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