# Category Archives: number theory

## The Sum of Cubes is the Square of the Sum

It’s fairly well-known, to those who know it, that . In other words, the square of the sum of the first n positive integers equals the sum of the cubes of the first n positive integers. It’s probably less well-known … Continue reading

## No Integer Solutions to a Mordell Equation

Equations of the form are called Mordell equations.  In this post we’re going to prove that the equation has no integer solutions, using (with one exception) nothing more complicated than congruences. Theorem: There are no integer solutions to the equation … Continue reading

## A Lesson on Converting Between Different Bases

We’re in the time of COVID-19, 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

## 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

## Arguments for 0.9999… Being Equal to 1

Recently I tried to explain to my 11-year-old 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

## 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

## 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

## 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

## Some Divisibility Properties of the Binomial Coefficients

Recently I read in Koshy’s book [1] on Catalan numbers some divisibility properties of the binomial coefficients I had not seen before.  Koshy credits them to Hermite.   They are particularly interesting to me because (as Koshy notes) some of the most famous … Continue reading