I am an amateur and have been doing some number theory for fun, so I apologize if my post is absolutely trivial:)
I have been playing with primality tests and I thought of the following method: Pick a number that you want to test for primality, say 13.List all whole number addends of the number as pairs:
1 12
2 11
3 10
4 9
5 8
6 7
If all pairs are relatively prime (no number greater than 1 can be "factored out"), the number is prime.
This is all pretty simple but I wanted to know where is this mentioned in the literature and where one could go from here to improve the algorithm or expand on this simple idea.
Thank you!
This is probably not mentioned in the literature, but you can prove it like this. Suppose that $n$ is your number and that $p$ is a prime dividing $n$. Let $q = n - p$, so that one line of your list will be $$ p + q = n.$$ As $p$ divides $p$ and $p$ divides $n$, we get that $p$ divides $q$. Thus a nontrivial divisor will lead to at least one line where all terms share a common factor.
More generally, if $a$ is any number that shares a nontrivial factor $p$ with $n$, and $a + b = n$, then $p$ must also divide $b$. From this point of view, to show a number is not prime in this method amounts to finding another number that shares a common factor with $n$. This is very similar to trial division --- just looking at all numbers less than $n$ for a common factor.
However you can give speedups analogous to those for trial division. It's only necessary to list those numbers $a \leq \sqrt{n}$ in your list to look for something with a common factor, for example.