# Pólya's conjecture

Pólya's conjecture is a hypothesis put forward by the Hungarian mathematician George Pólya
(1887–1985) in 1919. A positive integer is said to be of **even
type** if it factorizes into an even number of prime
numbers; otherwise it is said to be of **odd type**. For
example, 4, = 2 × 2, is of even type, whereas 18, = 2 × 3 ×
3, is of odd type. Let *O*(*n*) be the number of odd type
and *E*(*n*) be the number of even type integers in the first *n* integers. Pólya's conjecture says that *O*(*n*)
> *E*(*n*) for all *n* > 2. After the conjecture had
been checked for all values of *n* up to one million, many mathematicians
assumed it was probably true. However, in 1942 Ingham came up with an ingenious
method to show how a counterexample could be constructed, even though there
wasn't enough computing power around at the time to do the necessary calculations.
Twenty years later, R. S. Lehman ran Ingham's method on a computer to find
a counterexample to Pólya's conjecture at *n* = 906180359.