# number theory

## Finding the N-th digit of Pi

Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:Pi =...

## Large Counterexample

A positive integer is said to be of even type if its factorization into primes has an even number of primes. Otherwise it is...

## Sums of Two Squares

Which whole numbers are expressible as sums of two (integer) squares? Here’s a theorem that completely answers the question, due...

## Pythagorean Triples

Which triples of whole numbers {a, b, c} satisfy a2 + b2 = c2 ? Such triples are called Pythagorean triples because they are integer...

## Sums of Three and Four Squares

How many squares does it take to express every whole number as the sum of squares? We saw that two...

## Products of Sums of Two Squares

Here’s a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the...

Fibonacci numbers exhibit striking patterns. Here’s one that may not be so obvious, but is striking when you see it....

## Prime Number Theorem

Fix some number N. What fraction of the integers less than or equal to N are prime? Thinking about it,...