Noodling on Primes

As I am sure that every idea that I will ever have has already been thought of I thought it easiest to ask… Looking for the naming of names. My malformed thought…

Whole numbers in as their simplest idea are simply a composition of 1’s
\Bbb{N}=\{1,1+1,1+1+1, 1+1+...\}

With primes, we take one step back and say "All whole numbers that are only composed with 1 and themselves. Plus for fun, not 1.

\Bbb{P}=\{2,3,5,7,11,...\}

Somewhere along the line, we picked base 10 as a wheel. It could have been 12 or even 11 or 7. For ease of multiplication 12 would have been nice and for ease of memory 7 would have been more natural but we have fingers.

Back to the malformed thought. What would a wheel of primes look like? They are pseudo-random. There are definitely indefinite patterns (lol). We know they don’t end in \{ 2,4,5,6,8, 0 \} with the annoying exception of 2 and 5. So primes end \{1, 3, 7, 9\}.

Are there numbers that it makes sense to chop up a circle in to see patterns? Do different patterns appear depending on which prime you chop the circle up into.

Fermat’s little theorem , image

and then
Euler’s totient theorem, image

Wilson’s Therom

any number n is a prime number if, and only if, (n − 1)! + 1 is divisible by n.[1]

Prime Number Theorem.

Are there other neat things we can extract from this idea of prime modular arithmetic?

Not sure who has the next cool theorem I should be looking at with prime number relationships and what it might be called?

thanks,
r

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I don’t know anything about famous theorems since I never studied higher math myself.

As for using base 10, my guess for why we as humans do that is because we have 10 fingers. If we had 8 fingers we might use base 8 instead.

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