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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 .
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 ...
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 ...
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 ...
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.
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[2]: 63 It was Pólya who had the idea for a comprehensive problem book in analysis first, but he realised he would not be able complete it alone. He decided to write it with Szegő, who had been a friend of Pólya's since 1913, when the pair met in Budapest (at this time, Szegő was only 17, while Pólya was a postdoctoral researcher of 25).
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.