Pólya conjecture

The conjecture of Pólya referred a guess from the mathematical field of number theory. It says that the majority of the natural numbers are odd numbers with many prime factors up to an arbitrarily predetermined limit. The assumption was erected in 1919 by the Hungarian mathematician George Pólya, but refuted in 1958. The Pólya conjecture is an example that a mathematical statement that applies to numerous small numbers, but can be a total incorrectly due to ( relatively large ) counter-example.

Statement

Specifically, the conjecture of Pólya following: For a given natural number () to share the natural numbers from 1 up to and including into two sets, namely those with odd prime factors, many with just one hand and the many prime factors other. Then the first of these two sets contains at least as many numbers as the second. Here, multiple occurrences of prime factors are also counted several times accordingly so that for example, has an even number of factors, namely piece, whereas three, so it has many odd factors. The 1 has a single prime factor, that is an even number.

The presumption can be alternatively formulated with the help of summatorischen Liouville function, and then states that

Applies to all. This is positive if just has many prime factors, and negative if there are odd lots. The omega -function returns the number of prime factors of a natural number.

Refutation

In 1958, CB Haselgrove first gave a proof that the conjecture of Pólya 's wrong, by showing the existence of a counter-example with.

A first explicit counterexample, namely R. Sherman Lehman in 1960 indicated; actually the smallest counterexample found Minoru Tanaka 1980.

The conjecture of Pólya applies to most in the area 906150257-906488079 not. The Liouville function grows here on positive values ​​of up to 829 ( for ).