Question about primes

[Savage]

So: here is my understanding of a prime number.

Any integer number can be written in the form A = BC (Every number "A" can be written as the multiplication of 2 other number "B and C") where A, B, and C are integers.

Primes are special in that A (prime) can only be written as BC where B = A (or C = A) and and C = 1 (B = 1 if C = A).

Lastly, B =/= C (Making it impossible for 1 to be prime).

==> Please correct my understanding if this definition is incomplete / incorrect.

Question:

If we take the even naturals {2,4,6....} and throw a 1 in there {1,2,4,6...} would 6 be prime in this set?

[dragtom]

I'm not sure what the official definition is,
but your description results exactly in the prime numbers.

also, how would 6 be a prime in that set?
Every even number can be divided by 2, making 2 the only even prime number.

If you'd only dividing by the numbers in the list, then yes: 6 can only be written as 6*1...
but, as far as I know, that's no longer called a prime number.

[antiaverage]

Part of the definition of prime relies on using the full set of natural numbers.

Also, your reason for why 1 is not prime is inaccurate. We don't accept 1 as prime because of the Fundamental Theorem of Arithmetic which states that every positive whole number can be written as a unique product of primes.

[CuCN]

The part about B and C not being the same can be easily be avoided by defining a prime A as having exactly two values of B satisfying A=BC (where A, B, and C are all positive integers). For a prime, those two values are 1 and A. For 1, there is only one possible value (1) so it isn't prime.
For all integers, not just positive ones, this definition requires exactly 4 values of B instead, because of negatives.

The concept of a prime can be extended to commutative rings (http://en.wikipedia.org/wiki/Prime_element), but neither the even integers nor the even integers with 1 included are rings. The even integers alone do not have a multiplicative identity, and the even integers with 1 are not closed under addition.

[Savage]

The part about B and C not being the same can be easily be avoided by defining a prime A as having exactly two values of B satisfying A=BC (where A, B, and C are all positive integers). For a prime, those two values are 1 and A. For 1, there is only one possible value (1) so it isn't prime.
For all integers, not just positive ones, this definition requires exactly 4 values of B instead, because of negatives.

The concept of a prime can be extended to commutative rings (http://en.wikipedia.org/wiki/Prime_element), but neither the even integers nor the even integers with 1 included are rings. The even integers alone do not have a multiplicative identity, and the even integers with 1 are not closed under addition.

So, would 6 still be prime in your opinion since 6 can only be written as 6*1 in the set I created?

As you pointed out, the set is not closed under addition, so are you trying to say the definition of primes requires a set to be a ring as antiaverage pointed out?

[CuCN]

The concept of "prime" doesn't make sense for your set because it is not a ring. The "A=BC" definition only works for the full set of natural numbers (for the "exactly 2 values" version) or the full set of integers (for the "exactly 4 values" version).

[Savage]

I guess my question is "Why doesn't it make sense?" I am not sure if the answer requires are more precise definition or is this something mathematicians have just generally agreed on?

But if we are to stop this conversation at "Your set of numbers needs to be a ring before you can examine primes" I am fine with that.

[Zawadx]

A prime is a number which has no divisors other than 1 and itself.

Yes, 6 is an E-Prime (A prime in the set of even numbers). However you don't have to include 1 in there, because it still complies with the above definition. 2, 6, 10 and 30 are all e-primes based off the definition.

The reason people are arguing with you is that the Fundamental theorem of Arithmetic doesn't apply to your set (60=2*30=6*10). So your definition is wrong for that.

[Savage]

A prime is a number which has no divisors other than 1 and itself.

Yes, 6 is an E-Prime (A prime in the set of even numbers). However you don't have to include 1 in there, because it still complies with the above definition. 2, 6, 10 and 30 are all e-primes based off the definition.

The reason people are arguing with you is that the Fundamental theorem of Arithmetic doesn't apply to your set (60=2*30=6*10). So your definition is wrong for that.

(side note: no primes exist in the set of natural numbers)
[Assuming positive numbers only]
My definition works for the set of even numbers including 1, because if we don't include 1, (1) your definition fails (you stated 1 must be in the set, which is not in this set), (2) primes do not exist in the set of evens.

Now, again, this would slightly break the fundamental theorem of arithmetic because of unique factorizations as you put it: Example: 60 = 2*30 or 6*10.

==> Even if the factorizations are not unique, my real concern has to do with "who cares if my set is not a ring?" --> how does that affect primality?

[Sqantic Pilau]

Lastly, B =/= C (Making it impossible for 1 to be prime).

This can not form  any sensible definition of a prime otherwise squares of actual primes would also be prime (for 4;  B=C=2 and for 9 B=C=3 and for 25 B=C=5). Whereas the squares of these "new/extra" primes wouldn't be prime eg 91=27.3 etc.
Quite a mess.