Really the definition of each of these is well known.
The questions for debate here are twofold.
The first if primes may only refer to natural numbers but not integers.
If integers there is no unique factorization.
ie. -15= (-3)5 = 3(-5)
Or 15 = -3 (-5) =( 3)( 5)
But this is just one issue, does it prevent one from defining primes over integers?
Could one say within absolute value, it does not matter p from -p?
The second issue here is a relation or theorem between the two concepts.(prime and coprime)
Let us have it this way
Comon factor implies nonprime
Therefore since:
if A implies B
Not B implies not A
Conclusion
Prime implies a pair of coprime.
If
p=a+b
then a and b are coprime for any reasonable partition. Over natural numbers. 1 excluded from a,b
EX
7=3+4=2+5
13=6+7=5+8=4+7=3+8=2+9
All these relatively prime!
SLimilar result obtainied for a,b over integers?
There are however nonprimes having such expression
9=5+4
However not all partitions of 9 are coprime.
ie. 9=3+6
but all partitions of primes would be.
Thus it seems that a possible definition of prime is expressible as the sum of two elements over ALL such partitions.
One has also
13=14-1= 15-2 =16-3 =...
also all coprime except 13=26-13 which is also excluded as involving a prime or factor thereof; also excluding 1 as a factor.
What else could be done or proved with this?
An upper limit (p+1)/2 is needed also for ntural numbers a and b, so as to prevent duplication.
So new definition of prime p
A natural number prime p is such that if
p=a+b
with a and b natural numbers in the range (2, (p+1)/2) then for all partitions with this restriction, a and b are always relative prime.
Proof is simple, assume not true, arrive at contradiction.
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