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, and proved false in 1958 by C. Brian Haselgrove.
Pólya conjecture states that for any n (> 1), if the natural numbers 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 the requisite number of times—thus 18 = 21 × 32 has 1 + 2 = 3 prime factors i.e. an odd number, while 60 = 22 × 3 × 5 has 4 prime factors, i.e. an even number.)
Equivalently, it can be stated in terms of the summatory Liouville function, with 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.
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.
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