My Math Forum Goldbach Theorem

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 January 7th, 2013, 06:36 AM #1 Newbie   Joined: Nov 2012 Posts: 24 Thanks: 0 Goldbach Theorem This is my another try for the theorem A formal proof will go as follows:- We take two odd primes p1 and p2 So that p1? p2 p1+ p2? 2p2 But p1 and p2 are odd primes and sum of 2 odd no is always even So p1+p2=2x 2x?2p2 x?p2 ...(1) Now if p1?p2 p1?x ...(2) combining (1) and (2) p1?x?p2 Now p2 is the smaller prime and 3 is the smallest prime and there are infinitely many primes so greatest prime is infinite ? ? x ? 3 So x can be anything from 3 to infinite , which in turn makes 2x every even no starting from 6. But if sum of two primes is equal to 2x then x can also be equal to 2 x belongs to 2 to infinity , So 2x is every even no greater than 2. But 2x is the sum of two primes p1 and p2. Every even no greater than 2 is the sum of two primes. Second case can easily be proved by taking 3,5.. as the third prime.
 January 7th, 2013, 07:00 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Goldbach Theorem Once again, you've proved that the sum of two odd primes is even. This is not the same as proving that a given even number can be expressed as the sum of two primes.
 January 7th, 2013, 08:05 PM #3 Newbie   Joined: Nov 2012 Posts: 24 Thanks: 0 Re: Goldbach Theorem this time I proved that x can be anything from x can be anything from 2 to infinite , which in turn makes 2x every even no starting from 4 and 2x=p1+p2
 January 8th, 2013, 06:20 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Goldbach Theorem You can lead a horse to water....
 January 8th, 2013, 07:12 AM #5 Senior Member   Joined: Mar 2012 Posts: 572 Thanks: 26 Re: Goldbach Theorem Just to add, this might be where you are confusing yourself: "So p1+p2=2x" This means x is the midpoint between two numbers we already know are primes. For instance p1 = 7, p2 = 3, x = 5 "x can be anything from 3 to infinite" This only tells us that any number from 3 to infinity is halfway between a pair of integers, which might or might not be primes. (Actually it is halfway between a finite set of pairs of integers - for instance x = 5 is halfway between 1 and 9, 2 and 8, 3 and 7, 4 and 6.) "But if sum of two primes is equal to 2x then x can also be equal to 2 x belongs to 2 to infinity , So 2x is every even no greater than 2. But 2x is the sum of two primes p1 and p2." No. 2x as you defined it is the sum of two primes, because you defined it that way. This doesn't mean that any number 2x must be the sum of two primes.
 January 8th, 2013, 08:01 AM #6 Newbie   Joined: Nov 2012 Posts: 24 Thanks: 0 Re: Goldbach Theorem If p1+p2=2x. Then to prove the conjecture we have to prove that x can be any no from 2 to infinite. In my previous attempt I proved that sum of two primes will be an even no and every even no is of the form 2x.
January 8th, 2013, 08:37 AM   #7
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Re: Goldbach Theorem

Quote:
 Originally Posted by surya In my previous attempt I proved that sum of two primes will be an even no and every even no is of the form 2x.
The sum of any two odd primes is even. (2+3 is odd, and 2 and 3 are primes, but that's a minor detail.) But that isn't the same as Goldbach's conjecture.

Let me try this a different way. A Fermat prime is a prime of the form $2^n+1$ for some positive integer n. The Fermat primes are 3, 5, 17, 257, 65537 and possibly some others greater than $2^{2^{33}}.$ The sum of two Fermat primes is even, since all Fermat primes are, by their definition, odd. But not all even numbers can be represented as the sum of two Fermat primes. For example, 12 is not the sum of two Fermat primes: all Fermat primes other than 3 and 5 are greater than 12, and none of the sums 2+2, 2+4, and 4+4 are 12.

If your proof was valid it would also show that Goldbach's conjecture is true restricted to the Fermat primes, and that's obviously wrong. Do you understand now?

 January 8th, 2013, 09:00 AM #8 Newbie   Joined: Nov 2012 Posts: 24 Thanks: 0 Re: Goldbach Theorem Please see the proof again I said p1 and p2 are two odd primes
 January 8th, 2013, 11:43 AM #9 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Goldbach Theorem See paragraphs 2 and 3 in my above post.
 January 8th, 2013, 11:09 PM #10 Newbie   Joined: Nov 2012 Posts: 24 Thanks: 0 Re: Goldbach Theorem It works the other way round as 1) 2 is the only even prime. 2)Fermat primes are only a subset of universal set of odd primes. For ex 11+7=18 which is an even no and neither of them is a Fermat prime though p1 and p2 may be a Fermat prime but it is not necessary .

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