Category Archives: proof

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

Cassini’s identity

My previous post asked you to take any Fibonacci number, square it, and also multiply the two adjacent Fibonacci numbers, and see if a pattern emerged. Here’s a table I made for the first 6 Fibonacci numbers: (Hmm, the numbers … Continue reading

Posted in algebra, fibonacci, induction, pattern, proof, solutions | Tagged , , | 8 Comments

Triangular number equations via pictures: solutions

Here are some solutions to my previous post. However, they are almost certainly not the only solutions! If you have other cool ways to visualize any of these (or any other triangular number equations) feel free to post in the … Continue reading

Posted in arithmetic, pattern, pictures, proof, solutions | Tagged , , , | 10 Comments

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

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

Triangunit divisors and quadratic reciprocity

Recall that the triangunit numbers are defined as the numbers you get by appending the digit 1 to the end of triangular numbers. Put another way, where denotes the th triangular number, and the th triangunit number. The challenge, posed … Continue reading

Posted in arithmetic, modular arithmetic, number theory, primes, proof | Tagged , , , , | 5 Comments