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

Continue reading...# number theory

## 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...

Continue reading...## 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...

Continue reading...## Pythagorean Triples

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

Continue reading...## 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...

Continue reading...## 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...

Continue reading...## Fibonacci GCD’s, please

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

Continue reading...## Prime Number Theorem

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

Continue reading...## Sum of Prime Reciprocals

It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges. But what about...

Continue reading...## e is irrational

If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is 1/e =...

Continue reading...