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.