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Pólya conjecture
Disproved conjecture in number theory / From Wikipedia, the free encyclopedia
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In number theory, the Pólya conjecture (or Pólya's 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 set forth by the Hungarian mathematician George Pólya in 1919,[1] 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. For this reason, it is more accurately called "Pólya's problem".
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The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general,[2] providing an illustration of the strong law of small numbers.