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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 .
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.
Conjecture Field Comments Eponym(s) Cites 1/3–2/3 conjecture: order theory: n/a: 70 abc conjecture: number theory: ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1 ⇒Erdős–Woods conjecture, Fermat–Catalan conjecture Formulated by David Masser and Joseph Oesterlé. [1] Proof claimed in 2012 by Shinichi Mochizuki: n/a ...
Colin Brian Haselgrove (26 September 1926 – 27 May 1964) was an English mathematician who is best known for his disproof of the Pólya conjecture in 1958. [1] Haselgrove was educated at Blundell's School and from there won a scholarship to King's College, Cambridge. He obtained his Ph.D., which was supervised by Albert Ingham, from Cambridge ...
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Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 10 9. The green spike shows the function itself (not its negative) in the narrow region where the Pólya conjecture fails; the blue curve shows the oscillatory contribution of the first Riemann zero. Harmonic Summatory Liouville function T(n) up to n = 10 3
The Polya enumeration theorem translates the recursive structure of rooted ternary trees into a functional equation for the generating function F(t) of rooted ternary trees by number of nodes. This is achieved by "coloring" the three children with rooted ternary trees, weighted by node number, so that the color generating function is given by f ...
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 ...