Proof of the Infinitude of Twin Primes and all even numbered pairs of primes
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From: David R Tribble on 17 Jul 2010 21:27 Archimedes Plutonium wrote: > For years now I thought I had not delivered a proof of the Infinitude > of Twin Primes, that somehow I came up > short, but due to a email conversation, I realized that > all along I had proven the Infinitude of Twin, Quad, 6th primes > and all other even multiples Primes. > > Proof of the Infinitude of Twin Primes: > INDIRECT (contradiction) Method, Long-form; Infinitude of Twin Primes > > (1) Definition of prime as a positive integer divisible > only by itself and 1. > > (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S > Reason: definition of primes > > (3) Let us instead pick the numbers of primes as > the succession of 2,3,5,7,. . , p(n), p(n+2) where > the p(n) and p(n+2) are twin primes > > (4.0) Suppose twin primes are finite, then 2,3,5, ..,p_n , > p_n+2 is the complete series set > with p_n and p_n+2 the last and largest twin primes Reason: this is > the supposition step > > (4.1) Set S are the only primes that exist Reason: from step (4.0) > This is the step in which I hesitated in calling > my proof a genuine proof because I pictured larger regular primes > beyond the p_n+2, but that was superfluous > > (4.2) Form W+1 = (2x3x5x, ..,xp_n x p_n+2) + 1. > And form W-1 = (2x3x5x, ..,xp_n x p_n+2) - 1. > Reason: can always operate and > form a new number > > (4.3) Divide W+1 and W-1 successively by each prime of > 2,3,5,7,11,..p_n+2 and they all leave a remainder of 1. > Reason: unique prime factorization theorem W-1 is not necessarily prime. Consider 2 x 3 x 5 x 7 - 1 = 209 = 11 x 19.
From: Archimedes Plutonium on 18 Jul 2010 02:13 David R Tribble wrote: > Archimedes Plutonium wrote: > > For years now I thought I had not delivered a proof of the Infinitude > > of Twin Primes, that somehow I came up > > short, but due to a email conversation, I realized that > > all along I had proven the Infinitude of Twin, Quad, 6th primes > > and all other even multiples Primes. > > > > Proof of the Infinitude of Twin Primes: > > INDIRECT (contradiction) Method, Long-form; Infinitude of Twin Primes > > > > (1) Definition of prime as a positive integer divisible > > only by itself and 1. > > > > (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S > > Reason: definition of primes > > > > (3) Let us instead pick the numbers of primes as > > the succession of 2,3,5,7,. . , p(n), p(n+2) where > > the p(n) and p(n+2) are twin primes > > > > (4.0) Suppose twin primes are finite, then 2,3,5, ..,p_n , > > p_n+2 is the complete series set > > with p_n and p_n+2 the last and largest twin primes Reason: this is > > the supposition step > > > > (4.1) Set S are the only primes that exist Reason: from step (4.0) > > This is the step in which I hesitated in calling > > my proof a genuine proof because I pictured larger regular primes > > beyond the p_n+2, but that was superfluous > > > > (4.2) Form W+1 = (2x3x5x, ..,xp_n x p_n+2) + 1. > > And form W-1 = (2x3x5x, ..,xp_n x p_n+2) - 1. > > Reason: can always operate and > > form a new number > > > > (4.3) Divide W+1 and W-1 successively by each prime of > > 2,3,5,7,11,..p_n+2 and they all leave a remainder of 1. > > Reason: unique prime factorization theorem > > W-1 is not necessarily prime. > Consider 2 x 3 x 5 x 7 - 1 = 209 = 11 x 19. Yours is direct. Indirect, W-1 and W+1 are always necessary new primes, but do not feel bad because most mathematicians never got that correct either. That is why Twin Primes was never proved
From: sttscitrans on 18 Jul 2010 07:13
On 18 July, 02:27, David R Tribble <da...(a)tribble.com> wrote: > W-1 is not necessarily prime. > Consider 2 x 3 x 5 x 7 - 1 = 209 = 11 x 19. In the "indirect" method, the deranged AP mind has convinced itself that w+1 is "necessarily" prime. As AP is a complete loon, he can do this by blocking out, other correct alternatives. 1) Every n >1 has a unique prime factorization 2) GCD(n,n+1) =1 3) There is a last prime pn. w = PROD pi, i= 1 to n gcd(w,w+1) = 1 implies A) w+1 >1 has no prime divisors: contradicts 1) B) w+1 is not divisible by any prime <w+1, (1< pn <w <w+1) and so is a prime pj. If pj is a prime it must be one of the primes p1,.., pn these are ALL the primes assume to exist. gcd(w,w+1) = pj: contradiction C) w+1 has no prime divisors and so must equal 1 D) w+1 > pn implies w+1 is composite. => gcd(w,w+1) >1 :contradiction. The system of numbers described by 1),2) 3) cannot exist as it is self-contradictory. w+1 is prime yet has no prime divisors w+1 is >1 yet is equal to one w+1 is composite yet has no prime factors. Why AP thinks this tells him anything about the system of naturals which are consistent given primes are infinite is baffling in the extreme. He simply can't grasp that if it is not the case that the primes are finite in number then they must be infinite in number. |