Approach - primes vs non-primes
- if we were to define all integers as prime numbers, then it's certainly easy to prove that Goldbach's Conjecture is correct. we differentiate primes from non-primes (I can't remember what non-primes are called) because only primes have the sole factorization of the prime and 1, whereas non-primes have other factors.
So we start with a counting system, basically adding 1 to the previous number to get the next integer. from this viewpoint, all positive integers are alike.
then we discriminate between all positive integers which may be derived by adding 1 to the previous integer (primes) and those that have other factors (non-primes). So now we take a number system where all are the same and say, no, some are different.
What's confusing is that all integers are the same when we just add 1 over and over again to get each positive integer. but when we say we are going to derive those integers by multiplying 2 or more factors together to get that non-prime, then it's no longer the same as the primes - the numbers which can only be derived by counting.
So what i'm having trouble wrapping my head around, is we start from a counting or additive function and then try "subtract out" those integers which also have a multiplicative function and say, ok, how to we find the integers that are left, the primes?
How do we reconcile approaching integers from an additive function where they are all the same, to a multiplicative function where the very same set of integers are now split into 2 different classes? i realize multiplying is derived from adding, but they are not the same.
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