Author – Andrew J Frost 10/06/2020 REV TWO PnonP p3
OBSERVATION 3 – Patterns:- Primes and Non-Primes Endings…
In the table (as shown previously), the layout is composed of an integer column, followed by the product of the integer column and the 12x sequence. Then the uplift number (raised by) is added to produce the final resultant number.
The following spreadsheet example now shows results with any numbers ending in 5 sorted out and excluded from the spreadsheet.
This means that all non-prime odd-composite results are composed from factors that are either the number 7 or greater.
Please disregard the grey row showing the number 5.
Numbers that are magenta are squares or other powers; blue indicates odd-composite non-primes; primes are black.
Primes and Non-Primes Endings…
Ending in 1
- If the integer ends in 0 or 5 the resultant of x12 will end in 0, if then raised by the addition of 1 or 11 – all resulting primes and odd-composite numbers end in 1
- If the integer ends in 3 or 8 the resultant of x12 will end in 6, if then raised by the addition of 5 – all resulting primes and odd-composite numbers end in 1
- If the integer ends in 2 or 7 the resultant of x12 will end in 4, if then raised by the addition of 7 – all resulting primes and odd-composite numbers end in 1
Ending in 3
- If the integer ends in 1 or 6 the resultant of x12 will end in 2, if then raised by the addition of 1 or 11 – all resulting primes and odd-composite numbers end in 3
- If the integer ends in 4 or 9 the resultant of x12 will end in 8, if then raised by the addition of 5 – all resulting primes and odd-composite numbers end in 3
- If the integer ends in 3 or 8 the resultant of x12 will end in 6, if then raised by the addition of 7 – all resulting primes and odd-composite numbers end in 3
Ending in 7
- If the integer ends in 3 or 8 the resultant of x12 will end in 6, if then raised by the addition of 1 or 11 – all resulting primes and odd-composite numbers end in 7
- If the integer ends in 1 or 6 the resultant of x12 will end in 2, if then raised by the addition of 5 – all resulting primes and odd-composite numbers end in 7
- If the integer ends in 0 or 5 the resultant of x12 will end in 0, if then raised by the addition of 7 – all resulting primes and odd-composite numbers end in 7
Ending in 9
- If the integer ends in 4 or 9 the resultant of x12 will end in 8, if then raised by the addition of 1 or 11 – all resulting primes and odd-composite numbers end in 9
- If the integer ends in 2 or 7 the resultant of x12 will end in 4, if then raised by the addition of 5 – all resulting primes and odd-composite numbers end in 9
- If the integer ends in 1 or 6 the resultant of x12 will end in 2, if then raised by the addition of 7 – all resulting primes and odd-composite numbers end in 9
1, 3, 7, 9
This confirms the well known fact that any prime number can only end in 1, 3, 7, or 9.
In my spreadsheet, also confirmed is the fact that this rule equally applies to the odd-composite non-prime numbers that are developed as a Subset in the list.
