# Pólya conjecture

In number theory, the **Pólya conjecture** stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an *odd* number of prime factors. The conjecture was posited by the Hungarian mathematician George Pólya in 1919,[1] and proved false in 1958 by C. Brian Haselgrove.

The size of the smallest counterexample is often used to show how a conjecture can be true for many cases, and still be false,[2] providing an illustration for the strong law of small numbers.

## Statement

Pólya conjecture states that for any *n* (> 1), if we partition the natural numbers less than or equal to *n* (excluding 0) into those with an *odd* number of prime factors, and those with an *even* number of prime factors, then the former set has at least as many members as the latter set. (Repeated prime factors are counted the requisite number of times—thus 18 = 2^{1} × 3^{2} has 1 + 2 = 3 prime factors i.e. an odd number, while 60 = 2^{2} × 3 × 5 has 4 prime factors, i.e. an even number.)

Equivalently, it can be stated in terms of the summatory Liouville function, the conjecture being that

for all *n* > 1. Here, λ(*k*) = (−1)^{Ω(k)} is positive if the number of prime factors of the integer *k* is even, and is negative if it is odd. The big Omega function counts the total number of prime factors of an integer.

## Disproof

Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^{361}.[3]

An explicit counterexample, of *n* = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest counterexample is *n* = 906,150,257, found by Minoru Tanaka in 1980.[5]

The conjecture fails to hold for most values of *n* in the region of 906,150,257 ≤ *n* ≤ 906,488,079. In this region, the summatory Liouville function reaches a maximum value of 829 at *n* = 906,316,571.

## References

- Pólya, G. (1919). "Verschiedene Bemerkungen zur Zahlentheorie".
*Jahresbericht der Deutschen Mathematiker-Vereinigung*(in German).**28**: 31–40. JFM 47.0882.06. - Stein, Sherman K. (2010).
*Mathematics: The Man-Made Universe*. Courier Dover Publications. p. 483. ISBN 9780486404509.. - Haselgrove, C. B. (1958). "A disproof of a conjecture of Pólya".
*Mathematika*.**5**(02): 141–145. doi:10.1112/S0025579300001480. ISSN 0025-5793. MR 0104638. Zbl 0085.27102. - Lehman, R. S. (1960). "On Liouville's function".
*Mathematics of Computation*. Mathematics of Computation.**14**(72): 311–320. doi:10.2307/2003890. JSTOR 2003890. MR 0120198. - Tanaka, M. (1980). "A Numerical Investigation on Cumulative Sum of the Liouville Function".
*Tokyo Journal of Mathematics*.**3**(1): 187–189. doi:10.3836/tjm/1270216093. MR 0584557.