Prime quadruplet

Prime quadruplets consist of two twin prime pairs at a distance 4, ie four primes of the form p, p 2, p 6, p 8. In other words, there are exactly three numbers, which are all composed ( not prime ) between the two twin prime pairs. The smallest prime number quadruplets (5, 7, 11, 13), ( 11, 13, 17, 19 ), ( 101, 103, 107, 109 ) and ( 191, 193, 197, 199).

With one exception ( 5, 7, 11, 13) can each quadruple in the form ( 15n -4, 15n -2, 15n 2 15n 4) write. The number in the middle is therefore always divisible by 15 and the sum of the prime numbers of the quad is always divisible by 60. The numbers in the decimal system ends thus always at 1, 3, 7 and 9

Likewise, you can turn any quadruple either as ( 210n 101, 103 210n, 210n 107, 210n 109 ), ( 210n ± 11, ± 13 210n, 210n ± 17, ± 210 n 19) or ( ± 191 210n, 210n ± 193, ± 197 210n, 210n ± 199) write.

It is unknown whether there are infinitely many prime quadruplets. A prerequisite for infinitely many prime quadruplets is the existence of infinitely many twin primes; if this condition is satisfied is also not known.

Maynard and Tao showed in 2013 that there are infinitely many groups of four prime numbers whose elements are at most 25 million apart. Their proof uses methods from the work of Zhang's to twin primes. To prove the existence of infinitely many actual prime quadruplets, this limit should be reduced to 8.

According to the Hardy - Littlewood conjecture the number of prime quadruplets is less than x is asymptotically by the formula

( Sequence A061642 in OEIS ) was added.

The largest prime number Vierling has 3503 decimal places, was found in 2013 by Serge Batalov and is given by × 103490 2339662057597 d with d = 1, 3, 7, 9