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# Tag Archives: proof

## A computer-checked proof of the odd order theorem

Big news: a proof of the Feit-Thompson Theorem (also known as the “odd order theorem”) has been completely formalized and verified by a computer, using the Coq proof assistant! Wait, what? Huh? you’re probably thinking. Well, let me unpack that … Continue reading

## Fibonacci multiples, solution 1

In a previous post, I challenged you to prove If evenly divides , then evenly divides , where denotes the th Fibonacci number (). Here’s one fairly elementary proof (though it certainly has a few twists!). Pick some arbitrary and … Continue reading

Posted in fibonacci, modular arithmetic, number theory, pattern, pictures, proof, sequences
Tagged divisibility, fibonacci, proof, remainders
5 Comments

## Fun with repunit divisors: proofs

As promised, here are some solutions to the repunit puzzle posed in my previous post. (Stop reading now if you don’t want to see solutions yet!) Prove that every prime other than 2 or 5 is a divisor of some … Continue reading

Posted in iteration, modular arithmetic, number theory, pattern, primes, proof
Tagged divisibility, Fermat, prime, proof, repunit
1 Comment

## Book review: Roads to Infinity

What is infinity? What is proof? These are two of the biggest questions mathematicians have grappled with over the years. In this well-written and fascinating book, John Stillwell takes us on a tour through some of the answers to these … Continue reading

Posted in arithmetic, books, computation, induction, infinity, logic, proof, review
Tagged infinity, John Stillwell, proof, roads

## Triangular number equations via pictures

The other day I was fiddling around a bit with triangular numbers. By only drawing pictures I was able to come up with the following triangular number equations, where denotes the th triangular number (that is, the number of dots … Continue reading

## Irrationality of pi: the impossible integral

We’re getting close! Last time, we defined a new function and showed that and are both integers, and that . So, consider the following: The first step uses the product rule for differentiation (recalling that and ); the last step … Continue reading

Posted in famous numbers, proof
Tagged Fundamental Theorem of Calculus, integral, irrational, Ivan Niven, pi, proof
4 Comments

## Irrationality of pi: curiouser and curiouser

I’ve been remiss in posting here lately, which I will attribute to Christmas and New Year travelling and general craziness, and then starting a new semester craziness… but things have settled down a bit, so here we go again! Since … Continue reading

Posted in famous numbers, proof
Tagged derivatives, irrationality, Ivan Niven, pi, proof
10 Comments

## Irrationality of pi: derivatives of f

In my previous post in this series, we defined the function and showed that . Today we’ll show the surprising fact that, for every positive integer , although and are not necessarily zero, they are always integers. (The notation means … Continue reading

Posted in calculus, famous numbers, proof
Tagged derivatives, irrationality, Ivan Niven, pi, proof
10 Comments

## Irrationality of pi: the unpossible function

Recall from my last post what we are trying to accomplish: by assuming that is a rational number, we are going to define an unpossible function! So, without further ado: Suppose , where and are positive integers. Define the function … Continue reading

Posted in calculus, famous numbers, proof
Tagged irrational, Ivan Niven, pi, proof, symmetric
7 Comments

## Irrationality of pi

Everyone knows that —the ratio of any circle’s diameter to its circumference—is irrational, that is, cannot be written as a fraction . This also means that ‘s decimal expansion goes on forever and never repeats …but have you ever seen … Continue reading