Category Archives: proof

Book review: Fermat’s Enigma

Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical ProblemSimon Singh After having it recommended to me several times, I finally picked up this book when I happened to see it at our favorite local used bookstore. I … Continue reading

Posted in books, open problems, proof, review | Tagged , , , | 10 Comments

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

Posted in algebra, links, programming, proof | Tagged , , , , , , | 10 Comments

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 , , , | 5 Comments

Combinatorial proofs

Continuing from a previous post, we found that if we begin with th powers of consecutive integers and then repeatedly take successive differences, it seems like we always end up with the factorial of , that is, . We then … Continue reading

Posted in combinatorics, pictures, proof | Tagged , , | 12 Comments

A self-square number

[This is the seventh in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic … Continue reading

Posted in arithmetic, infinity, iteration, modular arithmetic, proof | Tagged , , , | 12 Comments

Fun with repunit divisors: more solutions

In Fun with repunit divisors I posed the following challenge: Prove that every prime other than 2 or 5 is a divisor of some repunit. In other words, if you make a list of the prime factorizations of repunits, every … Continue reading

Posted in arithmetic, iteration, modular arithmetic, number theory, primes, programming, proof, solutions | Tagged

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 , , , , | 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 , , ,

Collatz conjecture: an apology

In the comments on my previous post about Gerhard Opfer’s proposed proof of the Collatz conjecture, several different people expressed the opinion that my tone was rather arrogant, and I think they have a point. So: I apologize for my … Continue reading

Posted in meta, open problems, proof | 9 Comments

The Collatz conjecture is safe (for now)

A few days ago John Cook reported a draft paper claiming to solve the Collatz conjecture. Of course, since the Collatz conjecture is so simple to state, it constantly attracts tons of would-be solvers, and most of the purported “proofs” … Continue reading

Posted in open problems, proof | 43 Comments