Tag Archives: proof

Fun with repunit divisors: proofs

Posted on November 16, 2011 by

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 , , , , | 1 Comment

Book review: Roads to Infinity

Posted on July 5, 2011 by

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 , , ,

Triangular number equations via pictures

Posted on March 24, 2011 by

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 →

Posted in challenges, pictures, proof | Tagged , , , | 8 Comments

Irrationality of pi: the impossible integral

Posted on February 6, 2010 by

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 , , , , , | 4 Comments

Irrationality of pi: curiouser and curiouser

Posted on January 30, 2010 by

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 , , , , | 10 Comments

Irrationality of pi: derivatives of f

Posted on December 20, 2009 by

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 , , , , | 10 Comments

Irrationality of pi: the unpossible function

Posted on December 12, 2009 by

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 , , , , | 7 Comments

Irrationality of pi

Posted on December 7, 2009 by

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 →

Posted in calculus, famous numbers, proof | Tagged , , , | 14 Comments