## Pólya's conjecturePólya'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.## Related category• PRIME NUMBERS | |||||

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