proof of Goldbach conjecture
in [Math]
|
From: eestath on 1 Dec 2009 01:09 Goldbach conjecture states that every even integer greater then 4 is the sum of two primes Proof Theorem Golbach conjecture is true for every n>4 if the two prime numbers are different Proof /= (different from) Suppose that p/=q and that exist a k (positive integer) such that p+q/ =2k then we have: Is interesting to notice that p and q are odd. 2^(p+q)=2^(2k)=> 2*2^(p/q)/=2^(2k/q) Theorem 2^(p/q) is irrational number if q does not devide p ( in this case is allways irrational because p and q are different prime numbers) So 2*2(p/q) is irrational In order 2^(2k/q) to be different from 2*2^(p/q), 2^(2k/q) must be rational (it cannot be irrational from the assumption) So k must be devided by q or k=w*q (w>=1 a positive integer) We have a contradiction if k= w*q then we have: p+q/=2k=> p/=2k-q=> p/=2*w*q -q=> p/=q*(2w-1) q is odd from the assumption and 2w-1 is odd (the product of odd numbers is always odd) So p must be different from odd which contradicts the assumption. Q.E.D. If p=q Goldbach conjecture is true. Thus Goldbach conjecture is true for every number greater or equal to 4 Dimitris Stathopoulos Email: stathopouloscs(a)hotmail.com
From: Virgil on 1 Dec 2009 02:30 In article <bfe9e50f-05d8-41ac-87f7-cfd50b55a17f(a)f16g2000yqm.googlegroups.com>, eestath <stathopoulosee(a)gmail.com> wrote: > Goldbach conjecture states that every even integer greater then 4 is > the sum of two primes > > Proof > > Theorem > > Golbach conjecture is true for every n>4 if the two prime numbers are > different How about finding two different primes adding up to n = 6 ?
From: eestath on 1 Dec 2009 02:42 On Dec 1, 9:30 am, Virgil <Vir...(a)home.esc> wrote: > In article > <bfe9e50f-05d8-41ac-87f7-cfd50b55a...(a)f16g2000yqm.googlegroups.com>, > > eestath <stathopoulo...(a)gmail.com> wrote: > > Goldbach conjecture states that every even integer greater then 4 is > > the sum of two primes > > > Proof > > > Theorem > > > Golbach conjecture is true for every n>4 if the two prime numbers are > > different > > How about finding two different primes adding up to n = 6 ? if p=q is Conjecture is sutisfied...
From: eestath on 1 Dec 2009 02:48 i stated that for the cases that p=q the conjecture is true find a true error!!!!!
From: Henry on 1 Dec 2009 03:03
On 1 Dec, 06:09, eestath <stathopoulo...(a)gmail.com> wrote: > Goldbach conjecture states that every even integer greater then 4 is > the sum of two primes > > Proof > > Theorem > > Golbach conjecture is true for every n>4 if the two prime numbers are > different > > Proof > /= (different from) > Suppose that p/=q and that exist a k (positive integer) such that p+q/ > =2k then we have: > Is interesting to notice that p and q are odd. > > 2^(p+q)=2^(2k)=> > 2*2^(p/q)/=2^(2k/q) > > Theorem 2^(p/q) is irrational number if q does not devide p ( in this > case is allways irrational because p and q are different prime > numbers) > > So 2*2(p/q) is irrational > > In order 2^(2k/q) to be different from 2*2^(p/q), 2^(2k/q) must be > rational (it cannot be irrational from the assumption) Why not? there is more than one irrational number > > So k must be devided by q or > k=w*q (w>=1 a positive integer) > > We have a contradiction > > if k= w*q then we have: > > p+q/=2k=> > p/=2k-q=> > p/=2*w*q -q=> > p/=q*(2w-1) > > q is odd from the assumption and 2w-1 is odd (the product of odd > numbers is always odd) > So p must be different from odd which contradicts the assumption. Why not? There is more than one odd number. > Q.E.D. > > If p=q Goldbach conjecture is true. > > Thus Goldbach conjecture is true for every number greater or equal to > 4 > > Dimitris Stathopoulos > Email: stathopoulo...(a)hotmail.com |