Author Part 2 – Andrew J Frost 03/11/2021 REV TWO PnonP p21
Positions of Prime Factor Patterns in Odd-Composite Squares and other Exponents/Powers – Using Gaps – Andy Frost 2021/10/19
Another Look
Taking another look at the spreadsheet; there is a further pattern utilising the gap sequence 6, 4, 2, 4, 2, 4, 6, 2… at least this appears to be the case initially.
This new pattern is using the previously established gaps sequence occurring between the factors of the odd-composite numbers. But specifically the number is the one assigned to the squared factors in the odd-composite subset list.
As can be seen from the above spreadsheet image there is a regularity to the perfect squares of the prime factors of the special odd-composite numbers in the list. Also: –
Formula Illustrating the Gap Pattern Between Factors
As before stated in P&NonP – 20 – (page1) – the formula for this calculation is: – (prime factor1 x prime factor2 = non-prime odd-composite) – (previous odd-composite)/prime factor = resultant gap number. Prime factors 1&2 are the primes prime-factors under consideration; non-prime odd-composite is literally the previous odd-composite number in the generated specific number line list. “Prime” is the prime in the extracted list under consideration.
Squares and Exponents
The spreadsheet file that shows the squares and exponents as extracted values from the fully formulated spreadsheet is shown as a link below. The file is stored in my MediaFire account in folder “Spreadsheet – PrimeFactors – Exponents”, file name is “PnonP-P-FactorExponents-20211103.ods”.
https://www.mediafire.com/file/ikjjsz2tvzrdd7y/PnonP-P-FactorsExponents-20211103.ods/file
The image below contains the information shown in an extract from the above named spreadsheet. The prime factors of the odd-composite numbers sorted in this list. The last column in the image shows the exponent pattern with the previous column showing the power resultant number.
The Squares and Other Exponents
In the extracted image above are shown the prime factors of the odd-composite numbers with exact power values. These start with the squares such as 7 x 7 =49. Looking at the column headed “integr gap” it can be seen that any number that is formed by squaring the the prime factors is produced by an integer number that always ends in “0” or “4”. Also this number is always created by using the uplift of “1” as shown in the column headed “+1 5 7 11”; that is “integer x 12 + 1“.
The factors of the numbers are shown in the columns following on. If we then go to the last two columns; in the first column of these two, headed “power result”, is the square of the odd-composite non-prime under consideration. In the last column headed “exponent” is the “power” from which the “power result” has been created. Moving back to the column headed “div1 lowest prime factor” is shown exactly that. So for a factor of 7 we have 49 in the “power result” column.
The formula in the “exponent” column has identified the “power” or “exponent” by which the factor produces the “power result”. This illustrates that the squares and squares only are produced by an uplift of 1. If we look at an exponent of 3, there is an uplift of 5, 7, or 11.
If we then look at an exponent of 4, which is essentially an exponent of 2 x 2, there is an uplift of 1 again. It is clear that an odd number of factors will always have an odd numbered exponent, while an even number of factors will always have an even numbered exponent. Also for an even number of factors the power result always, as a number, ends in 1 or 9; while for an odd number of factors the power result ends in 1, 3, 7, or 9.
This result affects the gap sequence.
Please now download the associated spreadsheet (link shown above) if you have not done so already. This spreadsheet is small and is a “sort” on a much larger spreadsheet. This spreadsheet has the formulas removed and data retained to enable the “sort” on the value of the exponents.
Looking at this information, scroll over to the the part of the spreadsheet that checks the values of the gap separation of the exponent values. Column 73 onwards. Those values shown in magenta are the squares, cubes, etc., in other words where the factor is always the same factor.
The former gap pattern occurring elsewhere, i.e. 6,4,2,4,2,4,6,2…. also occurs here. The separation values within the column list of each factors are extracted as the same pattern. However moving down through the magenta values suddenly we come to two values of 4 one after the other (bordered in a green line), thus breaking the pattern.
On closer examination however it can be seen that the pattern is not broken. The pattern is identifying squares only. The other listed entries are results of exponents to higher odd powers of a factor.
Again Repeating the Formula Illustrating the Gap Pattern Between Factors
The formula for this calculation is: – (prime factor1 x prime factor2 = non-prime odd-composite) – (previous odd-composite)/prime factor = resultant gap number. Prime factors 1&2 are the primes prime-factors under consideration. Non-prime odd-composite is literally the previous odd-composite number in the generated specific number line list. “Prime” is the prime in the extracted list under consideration.
To Summarise on Formulas and Patterns
This formula shows the gap pattern occurring throughout the distribution of the factors of Non-Prime Odd-Composite numbers in this Set of Non-Prime Odd-Composite numbers, which is contained within the Set Saf. But it also shows that the same pattern describes the relationship between squares of prime numbers.
Full Larger Spreadsheet: The file is stored in my MediaFire account in folder “Spreadsheet – PrimeFactors – Exponents”, file name is “PnonP-P-Factor-OF-20211124.ods”.
https://www.mediafire.com/file/o4u2xx32t63y333/P%2526nonP-Factors-OF-20211124.ods/file
This spreadsheet contains the factors of non-prime odd-composites as well as primes down to row 6249 (prime 23399).
