## Monday, January 16, 2017

### Twin Primes (1)

I have mentioned before how the sum of all twin primes (excepting the first pair of 3 and 5) appears to be divisible by 12.

I then decided to look at the pattern of results obtained when the sum of successive pairs is divided by 12 and found an unexpectedly interesting pattern.

So when we divide the sum of 5 + 7 by 12, we obtain 1.

Then for the next successive pairs,

(11 + 13)/12     =  2
(17 + 19)/12     =  3
(29 + 31)/12     =  5
(41 + 43)/12     =  7

So there is a perfect matching here with the first four prime numbers. Though this pure pattern breaks down with the next few results, there is still a remarkably close relationship with the sequence of primes (with a difference of only 1 from the correct results in the sequence in evidence. So

(59 + 61)/12     = 10 (as opposed to 11)
(71 + 73)/12     = 12 (as opposed to 13)
(101 + 103)/12 = 17
(107 + 109)/12 = 18 (as opposed to 19)
(137 + 139)/12 = 23

Now with the next pair of twin primes the pattern breaks down somewhat with
(149 + 151)/12 = 25. So this value seems to be sticking out on its own (as compared to the next prime 29 . Then this is followed by,

(179 + 181)/12 = 30 (as opposed to 31) and
(191 + 193)/12 = 32
(197 + 199)/12 = 33

So we seem to have two additional results, here not matched by a corresponding prime before the next
(227 + 229)/12 = 38 (as opposed to 37) and
(239 + 241)/12 = 40 (as opposed to 41)

Then the next,

(269 + 271)/12 = 45 (which again appears as an additional results before the next,

(281 + 283)/12 = 47 and
(311 + 313)/12 = 52 (as opposed to 53).

Then we seem to bypass the primes i.e. 59, 61 and 67 before the nest two results

(419 + 421)/12 = 70 (as opposed to 71) and
(431 + 433)/12 = 72 (as opposed to 73)

And the next we have

(461 + 463) 12 = 77 (as opposed to 79)

So here the difference is 2, and the result is indicative of the fact that close relationship with the sequence of primes is slowly breaking down. So the remaining results up to 100 are

(521 + 523)/12 = 87
(569 + 571)/12 = 95

So again we seem to be missing a value for 83 while 87 and 95 differ from the next two primes 89 and 97 by 2 respectively.

However, it still seems remarkable that the results when dividing the sums of successive pairs of twin primes, should match the corresponding sequence of prime numbers so closely.