number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s less than any given number have an ''odd'' number of

prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s. The

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

was set forth by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. Though mathematicians typically refer to this statement as the Pólya conjecture, Pólya never actually conjectured that the statement was true; rather, he showed that the truth of the statement would imply the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pu ...

. For this reason, it is more accurately called "Pólya's problem". The size of the smallest

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...

is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general, providing an illustration of the strong law of small numbers.

Statement

The Pólya conjecture states that for any ''n'' > 1, if the

natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

less than or equal to ''n'' (excluding 0) are partitioned 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 repeatedly; for instance, we say that 18 = 2 × 3 × 3 has an odd number of prime factors, while 60 = 2 × 2 × 3 × 5 has an even number of prime factors. Equivalently, it can be stated in terms of the summatory Liouville function, with the conjecture being that :

L(n) = \sum_^n \lambda(k) \leq 0

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

The 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 × 10³⁶¹. An explicit counterexample, of ''n'' = 906,180,359 was given by R. Sherman Lehman in 1960; the smallest counterexample is ''n'' = 906,150,257, found by Minoru Tanaka in 1980. 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

External links

* {{DEFAULTSORT:Polya conjecture Disproved conjectures Conjectures about prime numbers