Author – Andrew J Frost 10/06/2020 REV TWO PnonP p13
OBSERVATION 13 – Mirror Patterns in the Squares of Primes
Today (2018-12-30, first noted) is a record of my efforts to appreciate the significance of the repeating mirror pattern in the ending 2 digits of the squares produced when my spreadsheet formula (Eq2) has values where: – “intcol” = “introw”, i.e. a=b. (break line | is the mirror point.)
The squares are produced as a descending sloping diagonal when using formula Eq2 in the spreadsheet.
The above image shows the squares in the diagonal line when a = b in Eq2. The numbers of interest are bordered by a magenta line. Numbers with a green background show the prime as a prime factor which is squared to give a non-prime odd-composite. Numbers with a pink background show initially factors which are non-prime odd-composites and then above this is shown the actual prime factors of the number. In the squares prime factors are formed by two sets of primes as prime factors.
Examples of this are: –
- 2401 with prime factors of 74 i.e. 7 x 7 x 7 x 7
- 5929 with prime factors of 72 x 112 i.e. 7 x 7 x 11 x 11
- etc. as shown below –
Mirroring
The mirroring sequence is shown in the last two digits of these squares as follows (the mirror point is noted by |n| where ‘n’ is an id number for the mirror point): –
Starting with these ending numbers from 1, as: |0| 01 49 21 69 89 61 29 41 61 69 81 49 09 01 09 81 21 89 41 29 |1|; they occur
in these squares: – |0| 01, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329 |1| >>…(with 2401 illustrated as the first odd-composite non-prime number colour red composed of the product of prime factors which only occur as primes in my list Set Saf)
and then they are mirrored as: – |1| 29 41 89 21 81 09 01 09 49 81 69 61 41 29 61 89 69 21 49 01 |2|
in these squares of numbers: – ** |1| >>5929, 6241, 6889, 7921, 8281, 9401, 10201, 10609, 11449, 11881, 12769, 14161, 14641, 16129, 17161, 17689, 18769, 19321, 20449, 22201 |2| >>…
and then we repeat the mirror sequence: – |2| 01 49 21 69 89 61 29 41 61 69 81 49 09 01 09 81 21 89 41 29 |3|
in these squares: – ** >> |2| 22801, 24649, 25921, 26569, 27889, 28561, 29929, 32041, 32761, 34969, 36481, 37249, 38809, 39601, 41209, 43681, 44521, 47089, 48841, 49729 |3| … etc. as below: –
and then mirrored again as you descend the slope of the spreadsheet: – |3| 29 41 89 21 81 09 01 09 49 81 69 61 41 29 61 89 69 21 49 01 |4| 01 49 21 69 89 61 29 41 61 69 81 49 09 01 09 81 21 89 41 29 |5| 29 41 89 21 81 09 01 09 49 81 69 61 41 29 61 89 69 21 49 01 |6| 01 49 21 69 89 61 29 41 61 69 81 49 09 01 09 81 21 89 41 29 |7| 29 41 89 21 81 09 01 09 49 81 69 61 41 29 61 89 69 21 49 01 |8| 01 49 21 69 89 61 29 41 61 69 81 49 09 01 09 81 21 89 41 29 |9| 29 41 89 21 81 09 01 09 49 81 69 61 41 29 61 89 69 21 49 01 |10| 01 49 21 69 89 61 29 41 61 69 81 49 09 01 09 81 21 89 41 29 |11| 29 41 89 21 81 09 01 09 49 81 69 61 41 29 61 89 69 21 49 01 |12| 01 49 21 69 89 61 29 41 61 …etc.
These numbers occur in this mirrored sequence when div5 and div3 integers have been hidden or deleted from the spreadsheet view. The sequence is of 20 numbers mirrored by the same 20 numbers making a sequence 40 numbers long which I am assuming will repeat ad infinitum.
Odd-composite, Non-prime Squares
However note that the odd-composite non-prime squares have not been deleted from the sequence and that they occur in the normal seemingly random pattern. I have marked these square numbers in red in the text above. These square numbers have squares whose square-root number itself has prime factors, i.e. the square-root is composed of primes as prime factors from Set S(af). (see spreadsheet diagonal).
The last two numbers of the odd-composite non-prime squares are listed below as a sequence: –
01, |1| 29, 81, 61, 41, 89, 49, |2| 21, 61, 69, 09, 81, 89, 41, |3| 09, 09, 81, 69, 21, 01, |4| 01, 61, 29, 41, 81, 49, 21, 41, |5| 29, 81, 09, 49, 69, 29, 69, |6| 01, 61, 29, 61, 49, 09, 21, 89, |7| 29, 41, 89, 21, 01, 09, 81, 61, 89, 21, |8| 21, 29, 41, 69, 01, 89, 41, |9| 41, 21, 09, 09, 49, 69, 41, 61 69, 01, |10| 69, 89, 41, 61, 81, 49, 01, 09, 89, |11| 89, 81, 09, 01, 61, 41, 21, 49, 01, |12|.01, 69, 29, 61…etc….
The break line |n| corresponds to the break line in the square number sequences shown above. As noted this starts with the 01 from the square 2401 and carries on from there. There is no mirroring or apparent regularity to this series.
The amount of ending numbers of odd-composite non-prime squares in each part of this series are as follows: – |0| 1 |1| 6 |2| 7 |3| 6 |4| 8 |5| 7 |6| 8 |7| 9 |8| 7 |9| 10 |10| 9 |11| 9 |12| …….
Mirroring in the single ending digit
Looking at the first two rows of the mirrored sequence of 40 numbers, we can see that it is made up from five-plus mirroring sub-sequences of the ending digit; each mirror sequence of 8 digits contains a mirroring sequence of 4 digits.
|8 1,9,1,9 |4 9,1,9,1 ||8 1,9,1,9 |4 9,1,9,1 ||8 1,9,1,9 |4 9,1,9,1 ||8 1,9,1,9 |4 9,1,9,1 ||8 1,9,1,9 |4 9,1,9,1 ||8…
This means that an even number of mirrored sequences will occur in 80 numbers, i.e. 10 sequences within the 80 numbers of eight digits mirrored and mirrored at every four digits within the eight.
Going back to the series of numbers produced by the formula Eq2 in the spreadsheet, when a=b (the diagonal line): –
- – (Just briefly reflect back to my previous sequence (OBSERVATION 5 – Patterns:- The Last Two Ending Digits) – identifying the last two digits of numbers in my list/set ‘S(af)‘ as a repeating sequence of 80 numbers, composed of three sub-sequences; two of 26 numbers and one of 28, which are primes and non-prime odd-composites.)
