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,...