Tuesday, 11 January 2011

11# Twin Prime Symmetry

When I started looking at twin primes I got fascinated by the primorial length symmetrical patterns formed by composite numbers that are multiples of a limited set of factors (eg the pattern for multiples of 5 and 7 is symmetrical over a primorial length pattern of 210 numbers, and so is the pattern for twin prime candidates, pairs of 6N+/-1 numbers for which neither is a multiple of 5 and 7).

It can be hard to picture the symmetries I am talking about due to the length of series involved. Here is an image of the start, finish and centre of the 5/7/11 twin prime symmetry. You can see that the pairs struck out at 11 and 2299, 2101 and 209, and 2057 and 253 are the only pairs cast out by adding 11 to the sieve in this section, and that they are symmetrically arranged (as are all the other pairs cast out by 5 and 7).

Similarly you can see the pairs containing 1067 and 1243, and 1199 and 1111 are cast out by 11 and are symmetrical around the central point between 1151/1153 and 1157/1159.

Note that there is a gap of 2 spaces at the centre of the symmetry. Adding 11 to the symmetry has led to 30 pairs being cast out - 15 above the centre and 15 below.
These include the pair 11 and 13 which is of course a twin prime even though it is included in this symmetry.

135 of 385 twin pair candidates remain in this pattern from start to finish and the pattern will repeat from 2310 to 4620.

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