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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 .
[4]: 23–24 The specific topics treated bear witness to the special interests of Pólya (Descartes' rule of signs, Pólya's enumeration theorem), Szegö (polynomials, trigonometric polynomials, and his own work in orthogonal polynomials) and sometimes both (the zeros of polynomials and analytic functions, complex analysis in general).
Mumford conjecture: geometric invariant theory: Haboush's theorem: 1976: Kenneth Appel and Wolfgang Haken: Four color theorem: graph colouring: Traditionally called a "theorem", long before the proof. 1976: Daniel Quillen; and independently by Andrei Suslin: Serre's conjecture on projective modules: polynomial rings: Quillen–Suslin theorem ...
Polya begins Volume I with a discussion on induction, not mathematical induction, but as a way of guessing new results.He shows how the chance observations of a few results of the form 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, etc., may prompt a sharp mind to formulate the conjecture that every even number greater than 4 can be represented as the sum of two odd prime numbers.
The Pólya enumeration theorem, also known as the Redfield–Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by J. Howard Redfield in 1927.
Cramér's conjecture; Riemann hypothesis. Critical line theorem; Hilbert–Pólya conjecture; Generalized Riemann hypothesis; Mertens function, Mertens conjecture, Meissel–Mertens constant; De Bruijn–Newman constant; Dirichlet character; Dirichlet L-series. Siegel zero; Dirichlet's theorem on arithmetic progressions. Linnik's theorem ...
The earliest published statement of the conjecture seems to be in Montgomery (1973). [1] [2] David Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture due to a story told by Ernst Hellinger, a student of Hilbert, to André Weil. Hellinger said that Hilbert ...
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