Twin Primes and Ending Numbers Andy Frost 2020-06-03
If you build a numerical list such that no prime is missed (except for 2, 3, and 5) and this list will include 1 and certain odd-composite numbers (shown red), you will get the following, “My List”: –
1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 217, 221 …….
This list can be formed by using all integers in the number line starting with 0 and in turn multiplying them by 12 and then adding the sequence 1, or 5, or 7, or 11 in succession per integer (1, 5, 7, 11 being the four primes below twelve when 1 is considered as prime).
The list as created will include numbers ending in 5. Once the list is created, remove by sieving all those numbers ending in 5.
So where n is an integer in the number line (including and starting with 0);
then 12n + {1,5,7,11} – (sieve of 5) = my list Set Saf, mentioned above. This is a Set of numbers that has two obvious subsets of primes and odd-composite non-prime numbers. Where the odd-composite numbers occur, I have shown them in red above.
– – These lists can also be referenced and researched on OEIS. A038510 and A070884.
However, the item of interest here is the ending digit in each number in my list. These numbers are a regular and repeating pattern as long as the specifically created list is used. (The ending digits of odd numbered primes 1,3,7, and 9 has been known for a long time).
The Patterns
My pattern is: – 1, 7, 1, 3, 7, 9, 3, 9 for primes and odd-composites included. This pattern does not change and repeats ad infinitum.
Remembering that these ending digits represent a full number, we can conclude the following: –
1 to 7 is a gap of 6, 7 to 1 is a gap of 4, 1 to 3 is a gap of 2, 3 to 7 is a gap of 4, 7 to 9 is a gap of 2, 9 to 3 is a gap of 4, 3 to 9 is a gap of 6, 9 to 1 is a gap of 2.
So in any sequence of 8 numbers, each having the ending numbers shown in the pattern above, there can only be a maximum of three sets of twin primes (gap of 2), that is six prime numbers in all. Now taking into consideration that some of the numbers in the list are odd-composite numbers, should they occur in the sequence of eight ending numbers shown, then that will reduce the frequency of the twin primes accordingly. We are not interested in a prime twinning with an odd-composite, even though the pattern is maintained.
As we go up the number line, the frequency of odd-composites increases, and the size of the gap between primes caused by odd-composites increases.
To look at this another way, create a list of all the numbers in sequence that only end with 1’s, 3’s, 7’s, and 9’s, and then strike out the non-primes to show single primes and twins, then making the twins green just to highlight them: –
01, 07, 09, 11, 13,| 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103,| 107, 109, 111, 113, 117, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141….
Twin Primes and Twin Prime Pairs
So this shows up that there are just 9 green twin primes in the above small selection of numbers. For a double twin (two twins in close proximity – lets call them a “twin-pair”), the sequence of ending-numbers has to be 1, 3, 7, 9 (the middle four ending-numbers shown in the eight digit pattern above). The requirement for a twin-pair therefore is consistently that the gap between them is 4, and also it has to be ending numbers of the above sequence 1, 3, 7, 9, only, and in that order; the gap of 4 only occurring of course between 3 and 7.
What is the frequency of twin-pairs? I can say that in my list, built the way I have described, that is, in 50192 numbers from 1 to 188219, consisting of primes and odd-composite gap numbers; there are only 51 twin-pairs. There are also 2062 twin primes, 8190 twinned odd-composite numbers, 7712 single prime numbers and 3836 single odd-composite numbers.
In this list of 50192 numbers, the pattern 1, 7, 1, 3, 7, 9, 3, 9 never fails, and I am sure is infinitely repeating.
