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, â€¦ 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, â€¦ 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 true, 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.