Abstract
With one exception, namely {2, 3, 5, 7}, it is impossible to have four consecutive primes p1, p2, p3, p4 with p4 - p1 < 8. An interval of seven or less cannot contain more than three odd numbers unless one of them is a multiple of three. On the other hand, groups of four primes p, p + 2, p + 6, p + 8, usually called prime quadruplets, are fairly common. The first is {5, 7, 11, 13}, followed by {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829} and so on. Just as with prime twins, pairs of primes p, p + 2, it is conjectured that the sequence of prime quadruplets goes on for ever. Indeed, the apparently simpler prime twin conjecture is currently an unsolved problem of mathematics although in 1973, Jing-Run Chen proved a weaker form: There are infinitely many primes p such that p + 2 is either prime or the product of two primes (See Halberstam & Richert [1]). A similar result holds for quadruplets [1, Theorem 10.6]: There exist infinitely many primes p such that (p + 2) (p + 6) (p + 8) has at most 14 prime factors. The prime quadruplet conjecture would then follow if we could reduce ‘14’ to ‘3’ but this seems be a problem of extreme difficulty.
Published Version
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