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Twin Primes and Ending Numbers Andy Frost 2020-06-03 edit 2023-09-15 edit 2024-05-01

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 other 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 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 mentioned above. (sieve of 5 means numbers ending in 5)

Where odd-composite numbers occur, that is numbers that are not prime, I have shown them in red.

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, long time).

My pattern is: – 1, 7, 1, 3, 7, 9, 3, 9 for primes and odd-composites included. This pattern does not change.

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. This sequence of gaps can be expressed mathematically as ‘Modulo 10’.

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 (bold black) and the twins in green: –

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….

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 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.

Primes and Odd-Composite Gaps Andy Frost 2020-06-06 edit 2023-09-15 edit 2024-05-01

Using my list as previously described**, that is where n is any integer in the number line (including and starting with 0); then 12n + {1,5,7,11} – sieve of 5 = my list. This list is an infinite Set of numbers which I’m calling my special Set S_af, My List.

** In the previous document entitled “Twin Primes and Ending Numbers” it was illustrated that a twin prime is formed by two prime numbers with a separation gap of two, eg. 11,13 and 17,19.

Using the established ending number sequence, 1, 7, 1, 3, 7, 9, 3, 9, I concluded that ending numbers 1 to 7 produce a gap of 6, 7 to 1 produces a gap of 4, 1 to 3 produces a gap of 2, 3 to 7 produces a gap of 4, 7 to 9 produces a gap of 2, 9 to 3 produces a gap of 4, 3 to 9 produces a gap of 6, 9 to 1 is a gap of 2. This originates a gap pattern in the prime/non-prime, odd-composite list, Set S_af.

Gap Pattern in the prime/non-prime, odd-composite list, Set S_af.

The pattern is: – 6, 4, 2, 4, 2, 4, 6, 2, for primes and odd-composites on the number line in my list. This pattern also does not change. This can be checked in the list below which is the beginning of Set S_af : –

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 …….289 Red = odd-composite. Green = odd-composite true square.

The pattern repeats through the entire list continually and infinitely. The irregular spacing between primes is compensated for exactly by a similar but complimentary irregular spacing between odd-composite non-primes. The combination of the two sets of numbers from then on forms the rigidly regular repeating pattern within Set S_af.

The pattern indicates the importance of studying the primes together with the odd-composite non-primes as a single Set of numbers.

This same Gap pattern 6, 4, 2, 4, 2, 4, 6, 2, exists independently between Factors.

To illustrate this in the Set of numbers Set S_af; for any odd-composite number, choose any prime functioning as a factor of an odd-composite number in this Set. For the sake of ease of illustration I have worked with the first few prime-factors below. That is 7, 11, 13, 17, 19….. etc.

For prime 7, the first odd-composite that it occurs in is 49 (7² the square), of which 7 is the factor. To find the lowest denominator (i.e. gap), from this odd-composite number of 49, subtract 7 and the answer is 42, then divide by 7 and the answer is 6. The next odd-composite is 77, so doing the same again (i.e. subtract the previous odd-composite number 49), the answer is 28, divide by 7 and the result is 4. The next odd-composite is 91; subtract the previous odd composite 77, the answer is 14, divide by 7 and the result is 2. The next odd-composite is 119, repeating the process and the result is 4, for 133 the result is 2, for 161 the result is 4, for 161 the result is 6, for 203 the result is 2, 217 the result is 6. This process can be repeated and infinitely produces the pattern 6,4,2,4,2,4,6,2,|6……

For 11 the first odd-composite is 77, of which 11 and 7 are the factors. The pattern is repeating again, i.e. 77 – 11 = 66. 66 divided by 11 equals the resultant 6. So 121 – 77= 44 and 44/11= 4. And 143 – 121 = 22 and 22/11 = 2. And 187 – 143 = 44 and 44/11 = 4. ……etc.

So for the pattern 6, 4, 2, 4, 2, 4, 6, 2, …. for each odd-composite number in Set S_af as shown in the list above that is functioning as a factor, the factors on their own follow the same gap pattern. This can be seen in the following two pages.

The first illustration is an extract from the spreadsheet showing the results for primes 7 through to 19 when they are acting as factors for the odd-composite numbers in the list (primes have been sorted out from the list).

The second illustration shows an extract from the spreadsheet with the prime numbers (grey rows) included with odd-composite numbers in the series Set S_af . Also shown are the columns to the left which illustrate the rigid patterns associated with this series. That is “pattern 2”, ending numbers, “pattern 3”, gap numbers.

To summarise – there are two gap patterns which are the same but are not of the same circumstance, one is inclusive in the list of primes and odd-composite numbers, Set S_af, the other is solely occurring in the Subset of odd-composite numbers within Set S_af.

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Conclusions (some of many)

1. That as long as the list of numbers is formed as described to obtain Set S_af, all (odd) primes will be included to infinity along the Number Line.
2. Also a special sequence of odd-composites will be included to compliment the primes and make the list sequence complete.
3. That for each of these complimentary odd-composites its prime factors will have previously occurred as primes in the list.
4. That the gap sequence of all these numbers in the list is a constant and immutable sequence 6,4,2,4,2,4,6,2…repeating, also that in an immutably linked way the ending numbers of these numbers are a connected sequence of 1, 7, 1, 3, 7, 9, 3, 9….
5. That twin primes can only occur within the three ending number relationships of gap 2, underlined in the paragraph marked ***.
6. This relationship of gap 2 does not stop a prime and odd-composite twin, or an odd-composite to odd-composite twin gap occurring.

Observations in the spreadsheet extract above (just a few of many)

1. Pattern 6, the last two ending numbers
2. Pattern 4, the integer pattern required to multiply by 12 to obtain the number sequence
3. Pattern 5, is the ending number of the resultant of the pattern 4 process
4. Pattern 1, is the uplift sequence, the number required to be added to pattern 4 to result in the number under investigation
5. Pattern 2, is the ending number sequence
6. Pattern 3, is the gap sequence between the numbers in the numbers list, also between the factors of the odd-composites in the number list, and also between the numbers in the uplift sequence and the ending number sequence.