# Thread: Proof of twin prime conjecture

1. ## Proof of twin prime conjecture

PROOF OF THE TWIN PRIME CONJECTURE
The twin prime conjecture states that there are an infinite number of twin primes. Twin primes are a pair of primes that differ by 2 e.g. 11 and 13, 17 and 19, 29 and 31. Consider the sieve of Eratosthenes acting on the infinite number line. 2 makes the first move and eliminates all its infinite number of multiples, including itself, (i.e. the even numbers) from the infinite number line. 3 then makes the next move and eliminates all its infinite number of multiples (including itself) that have not already been eliminated by 2, from the infinite number line. At this point of the sieving process, it is not difficult to see that all the infinite number of integers left on the infinite number line (after 1 is removed also) are in pairs of the form 6n-1 and 6n+1, where n is some positive integer. Therefore odd primes are of the form 6n-1 or 6n+1. So, for particular values of n, twin primes, 6n-1 and 6n+1 will exist amongst the infinite integer pairs of the form 6n-1 and 6n+1 remaining on the infinite number line. So 5 makes the next move and eliminates all its infinite number of multiples (apart from itself) that have not already been eliminated by 2 and 3 i.e. (in ascending order) 5x5, 5x7, 5x11, 5x13, 5x17, 5x19, 5x23, 5x5x5, 5x29, 5x31, 5x5x7, 5x37, 5x41, 5x43, 5x47, 5x7x7, 5x53, 5x5x11,  7 then makes the next move and eliminates all its infinite number of multiples (apart from itself) that have not already been eliminated by 2, 3 and 5 i.e. 7x7, 7x11, 7x13,  11 then makes the next move and 13 the next after 11 and the process carries on ad infinitum.
So, following this process, the twin primes are simply those pairs of primes, 6n-1 and 6n+1 that escape the elimination process. A little thought will show that for the twin prime conjecture to be false, it takes one and only one special prime to eliminate (at some point in its elimination process) the remaining infinite number of 6n-1 and 6n+1 pairs of integers ahead of it on the infinite number line (i.e. those that have not already been eliminated by previous primes). Eliminating a pair of 6n-1 and 6n+1 integers simply requires that one of them is eliminated i.e. one of them is a multiple of this special prime. It will now be shown that it is impossible for any special prime to achieve this i.e. the twin prime conjecture is true.
In this part of the discussion, consider the infinite number line after 2, 3 and their infinite number of multiples have been eliminated (1 is removed also) as mentioned previously. Imagine a prime number hopping (infinitely) like a frog along this infinite number line consisting of integers of the form 6n-1 or 6n+1 during its turn in the sieving process, as it eliminates its multiples as described above e.g. 17 takes its first hop and eliminates 17 x 17 = 289 as it lands on it. 17 then takes its second hop and then eliminates 17 x 19 = 323 as it lands on it etc. A little thought will show that each and every one of the infinite number of multiples of 17 with prime factors of 5, 7, 11, 13 and 17 only will have a pair of un - eliminated 6n-1 and 6n+1 integers just before it (if the multiple of 17 is of the form 6n-1) or after it (if the multiple of 17 is of the form 6n+1). So as 17 is hopping infinitely along the infinite number line described, it will hop over these infinite number of 6n-1 and 6n+1 un-eliminated pairs. So this will be true for all primes greater than 3. So 37 will hop over un- eliminated pairs before or after the infinite number of multiples of 37 with prime factors of 5, 7, 11, 13, 17, 19, 23, 29, 31 and 37 only. So no such special prime exists that can eliminate all the remaining un-eliminated 6n-1 and 6n+1 pairs ahead of it at some point in its infinite hopping. So the twin prime conjecture is true i.e. there are an infinite number of twin primes.

2. ## Re: Proof of twin prime conjecture

The crucial point is the statement "So no such special prime exists that can eliminate all the remaining un-eliminated 6n-1 and 6n+1 pairs ahead of it at some point in its infinite hopping." But you have not proved that. You go up to 37 but what about primes larger than 37?

3. ## Re: Proof of twin prime conjecture

37 was just an example.

4. ## Re: Proof of twin prime conjecture Originally Posted by MrAwojobi 37 was just an example.
How high above 37 can you go then?

-Dan

5. ## Re: Proof of twin prime conjecture Originally Posted by MrAwojobi 37 was just an example.
That was exactly my point! You are giving examples, not a proof.

6. ## Re: Proof of twin prime conjecture

My explanation works for all primes apart from 2 and 3.

7. ## Re: Proof of twin prime conjecture Originally Posted by MrAwojobi My explanation works for all primes apart from 2 and 3.
But how do you know it's going to work for 7781183771 or 28081131187649? You gave an explanation, not a proof.

So how do you prove it?

-Dan

9. ## Re: Proof of twin prime conjecture Originally Posted by MrAwojobi Where is it proven? I didn't see any proof of that. I just saw an example using 37. Not a proof that an arbitrarily large prime can work the same way.

When I was in High School I proved the four color theorem. I had loads and loads of examples. But as it turned out I didn't prove the theorem for every possible construction.

You don't have a proof, just a likely argument.

-Dan

11. ## Re: Proof of twin prime conjecture

The fact that several people "haven't read it properly" indicates that either it is flawed or you haven't explained it properly!

12. ## Re: Proof of twin prime conjecture Originally Posted by MrAwojobi So, following this process, the twin primes are simply those pairs of primes, 6n-1 and 6n+1 that escape the elimination process. A little thought will show that for the twin prime conjecture to be false, it takes one and only one special prime to eliminate (at some point in its elimination process) the remaining infinite number of 6n-1 and 6n+1 pairs of integers ahead of it on the infinite number line (i.e. those that have not already been eliminated by previous primes). Eliminating a pair of 6n-1 and 6n+1 integers simply requires that one of them is eliminated i.e. one of them is a multiple of this special prime. It will now be shown that it is impossible for any special prime to achieve this i.e. the twin prime conjecture is true.
In this part of the discussion, consider the infinite number line after 2, 3 and their infinite number of multiples have been eliminated (1 is removed also) as mentioned previously. Imagine a prime number hopping (infinitely) like a frog along this infinite number line consisting of integers of the form 6n-1 or 6n+1 during its turn in the sieving process, as it eliminates its multiples as described above e.g. 17 takes its first hop and eliminates 17 x 17 = 289 as it lands on it. 17 then takes its second hop and then eliminates 17 x 19 = 323 as it lands on it etc. A little thought will show that each and every one of the infinite number of multiples of 17 with prime factors of 5, 7, 11, 13 and 17 only will have a pair of un - eliminated 6n-1 and 6n+1 integers just before it (if the multiple of 17 is of the form 6n-1) or after it (if the multiple of 17 is of the form 6n+1). So as 17 is hopping infinitely along the infinite number line described, it will hop over these infinite number of 6n-1 and 6n+1 un-eliminated pairs. So this will be true for all primes greater than 3. So 37 will hop over un- eliminated pairs before or after the infinite number of multiples of 37 with prime factors of 5, 7, 11, 13, 17, 19, 23, 29, 31 and 37 only. So no such special prime exists that can eliminate all the remaining un-eliminated 6n-1 and 6n+1 pairs ahead of it at some point in its infinite hopping. So the twin prime conjecture is true i.e. there are an infinite number of twin primes.
I did read this carefully. What you have here is a demonstration, not a proof. You have to show that your conjecture is correct, not just demonstrate it with a few primes.

-Dan

13. ## Re: Proof of twin prime conjecture

What I have here is a proof. Read it a few more times and maybe you will understand that this proof is a no brainer.

14. ## Re: Proof of twin prime conjecture Originally Posted by MrAwojobi What I have here is a proof. Read it a few more times and maybe you will understand that this proof is a no brainer.
"A little thought" does not make a proof. This has been pointed out several times and you don't seem to understand that. Supply the chain of reasoning between "a little thought" and "proves." Then come back and enlighten us.