Some clusters of prime numbers have special properties. Particularly the first one that we often learn are twin primes. Twin primes are primes that are close together as much as possible. Say, for instance, you have a prime like 17. The next number after an odd number is always an even number (here, it's 18), so it's not a prime (unless it's 2). Therefore we look at the number after that, which is 19. Now we have a pair of primes that differ by 2: 17 and 19. Are there other pairs of primes which have the same property? We start from the very start: 3 and 5. Then 5 and 7. Then 11 and 13. And so on. Pairs of primes which differ by 2 are deftly called twin primes. The pairs of twin primes less than 100 are:
3 and 5
5 and 7
11 and 13
17 and 19
29 and 31
41 and 43
59 and 61
71 and 73
Check again the twin primes above. Find out the number between each prime: 4, 6, 12, 18, 30, 42, 60, 72. Do you notice something? Except for the first, all of them are divisible by 6!
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Here is a quick proof, though, that except for the first pair, all integers between twin primes are divisible by 6: All positive integers greater than four can be expressed in exactly one of these forms:
6k-1, 6k, 6k+1, 6k+2, 6k+3, and 6k+4.
For all integer positive integer k (so we start k = 1, yielding 6 • 1 - 1 = 5), we see that:
1) 6k is not prime, because it is even and greater than 2;
2) 6k + 2 is not prime, because it is even and greater than 2;
3) 6k + 3 is not prime, because it is divisible by 3, like this: 6k + 3 = 3(2k + 1).
4) 6k + 4 is not prime, because it is even and greater than 2.
Therefore the possible candidates for twin primes (for all integer k greater than 1) consist of the pair 6k-1 and 6k+1. The integer between them is 6k, which is divisible by 6.As a corollary, for all pairs of twin primes greater than 4 (starting with 5 and 7), their sum is always divisible by 12.
5 + 7 = 12
11 + 13 = 24
17 + 19 = 36
29 + 31 = 60
41 + 43 = 84
59 + 61 = 120
71 + 73 = 144
(There are also some primes which are expressible in the form 6k+2 and 6k+3: by letting k = 0, we have 2 and 3. However, both are less than four, so they are immaterial to the above discussion.)
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The largest known twin primes, as of today, are 65516468355 • 2^333333 - 1 and 65516468355 • 2^333333 + 1. Both numbers were found out in 6 August 2009 and each has 100355 digits. As of today, there are 808,675,888,577,436 twin prime pairs less than 1,000,000,000,000,000,000 (10^18). All of these results make us wonder whether the list of twin primes never ends. Is there an infinitude of twin primes? Euclid, two thousand years ago, outlined a proof that the number of primes is infinite; today, the question regarding the infinitude of twin prime pairs is one of the most famous open questions in the theory of numbers.