Visualizing how prime numbers are revealed, and more
The four images below illustrate how a prime number is revealed by the lowest possible composite number above it. I have not come across this demonstration before. It captures how prime numbers naturally come about.
But this is not the only important lesson provided by the List of Prime and Composite Numbers 7 and above (LoPaC).
The List of Prime Numbers (LoP) is fully integrated within the LoPaC. The LoP is needed to determine if a large number is a prime or a composite, from number 7 up to the square root of that number.
How can you identify if a number is prime or not if that number is bigger than the square of the highest prime in the LoP? Or if you do not have a LoP?
LoPaC is the solution. Simply divide the large number with all the elements contained in LoPaC up to the square root of that number. The composites that reveal the primes are superfluous as they cannot divide the large number into 2 whole numbers with no remainder as only the prime numbers can.
LoPaC can then reveal if any given number, no matter how large, is a prime or a composite number, without having to know (calculate) if the elements within are prime or composite. This is inherently efficient.
As well, the LoPaC does not need to be kept into the computer’s memory as the LoP does because its elements can be calculated before each of the divisions necessary to determine if a given number is prime or not.
Details of the process will be forthcoming.
The following sentence in paragraph 5 is INCORRECT: "The composites that reveal the primes are superfluous as they cannot divide the large number into 2 whole numbers with no remainder as only the prime numbers can".
Composites in A007775 can also produce quotients with no remainders when dividing a given large number.
But that composite is composed of smaller primes, and the primes would have flagged the composite earlier, ending the divisions before the larger composite could have.
So I was right to think that the true role of the composites in the List of Primes and Composites was to accompany and render available the Complete List of Primes.