Category Archives: proof

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

P ≠ NP?

A few days ago, Vinay Deolalikar, a Principal Research Scientist at HP Labs, began circulating a draft of a paper entitled “P ≠ NP”. The mathematics and computer science communities immediately erupted in a frenzy of excitement and activity. The … Continue reading

Posted in computation, links, open problems, people, proof | Tagged , , , | 7 Comments

Irrationality of pi: the integral that wasn't

And now for the punchline! Today we’ll show that, for large enough values of , completing the proof of the irrationality of . First, let’s show that is positive when . We know that is positive for . But I … Continue reading

Posted in algebra, calculus, convergence, famous numbers, proof, trig | Tagged , , , , | 8 Comments

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 , , , , , | 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 , , , , | 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 , , , , | 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 , , , , | 7 Comments