Author – Andrew J Frost 10/06/2020 REV TWO PnonP p5
OBSERVATION 5 – Patterns:- The Last Two Ending Digits
If one looks at the last two digits of the prime and non-prime number series, there is also a repeating pattern in these numbers. I have included preceding zeros for the single digit numbers: –
The first 26 numbers in the spreadsheet table are: –
01, 07, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97.
The remaining numbers are a continuation series that can be identified as repeating as a line of 28 numbers and another line of 26 numbers, to make the overall list of 80 numbers. (Shown in 10 groups of 8 numbers)
The repeating pattern of the overall list of 80 numbers is as follows: –
01 07 11 13 17 19 23 29 – 31 37 41 43 47 49 53 59 – 61 67 71 73 77 79 83 89 – 91 97
01 03 07 09 13 19 – 21 27 31 33 37 39 43 49 – 51 57 61 63 67 69 73 79 – 81 87 91 93 97 99
03 09 – 11 17 21 23 27 29 33 39 – 41 47 51 53 57 59 63 69 – 71 77 81 83 87 89 93 99
Those numbers above shown in red are odd-composite non-primes. This sequence of 80 numbers then repeats ad infinitum.
Always remembering that these are endings and not necessarily full numbers, it is interesting to note that the odd-composite ending numbers in the first and last lines all appear in the middle line, except for 77. Also that the prime numbers that appear in the first line are repeated in the last two lines.
This is an intuitive indication of complementarity within the infinite series. But what about 77?
Full Numbers Containing the Two Ending Numbers
Regarding the actual full numbers; the second line is composed of numbers starting at 101 and ending at 199; the third line starts at 203 and ends at 299.
This means that the next iteration of the 80 number sequence starts at 301 and ends at 599, and so on.
There are 10 iterations of the sequence of single ending digits 1, 7, 1, 3, 7, 9, 3, 9, within this larger sequence of two ending numbers.
77 – Seventy Seven
In the last line, the number 77 is in fact the two ending numbers of 277 which is actually a prime. Primes are black and non-primes are red.
77 is an odd-composite non-prime, while 277 is prime.
Actual Numbers Line One – 01 07 11 13 17 19 23 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97
Where a dot occurs below; this indicates that either the number is prime (starting or all colour black) or is non-prime (starting or all colour red), or parts of it after the dot are individually prime or non-prime (black or red). If the number is all black without a dot it is solely prime; or all red with a dot, the number is solely non-prime. Solely in this usage means that both the two ending numbers and the full three digit number are individually prime, black, or non-prime, red.
Actual Numbers Line Two – 101 1.03 107 1.09 113 1.19 1.21 1.27 131 1.33 137 1.39 1.43 1.49 1.51 1.57 1.61 1.63 167 1.69 173 179 1.81 1.87 1.91 1.93 197 1.99
((Last Two Numbers, Line Two – 01 03 07 09 13 19 21 27 31 33 37 39 43 49 51 57 61 63 67 69 73 79 81 87 91 93 97 99))
Actual Numbers Line Three – 2.03 2.09 211 2.17 2.21 223 2.27 229 2.33 2.39 241 2.47 2.51 2.53 2.57 2.59 2.63 2.69 271 2.77 2.81 283 2.87 2.89 2.93 2.99
((Last Two Numbers, Line Three – 03 09 11 17 21 23 27 29 33 39 41 47 51 53 57 59 63 69 71 77 81 83 87 89 93 99)).
Conclusion on 77
The difference between 77 and other non-primes occurring in the “two ending digit list” is simple. All non-prime two digit ending numbers occurring in the list are divisible by factor 3, whereas 77 is the first number in the list divisible by factor 7 and a number greater than 7; which in this case is 11.
As initially stated, primes/(prime factors) 2, 3, and 5 are sieved out of my special list. The odd-composite non-primes in the list only use factors of 7 or other odd factors greater than 7.
