A prime number is a positive integer p that has exactly two unique divisors: 1 and p

No need for racism.

>do it without appealing to the principle of the excluded middle
Same proof but with binary products replaced by finite products, and using the definition in .

To start, 2 is not a zero or a unity.

Suppose 2 divides ab.

If 2 divides a and b, we are done.

WLOG, suppose 2 does not divide a. We have 2c = ab and a = 2k + 1 for some integers c and k. Therefore
2c = 2bk + b
and
2(c - bk) = b.
It follows that 2 must divide b.

QED.

Both definitions are equivalent

Finally, another proper proof.

>I thought mathematicians liked beauty and all that.

They do. And "beauty" is precisely why they excluded 1 from being a prime number.

It all comes down to the unique prime factorization theorem:

"All positive numbers have a unique ordered prime factorization."

For example, 6 has only one unique ordered prime factorization: 2 * 3. (The "ordered" requirement prevents you from claiming that 3 * 2 is also a factorization, because the prime factors must always be listed in non-descending order.)

If you allowed 1 to be considered "prime", then the unique prime factorization theorem would be false. Example:

6 = 2 * 3
6 = 1 * 2 * 3
6 = 1 * 1 * 2 * 3
etc.

So in order to expose the beauty and elegance of the unique ordered prime factorization theorem, mathematicians had to exclude 1 from being prime.

(This is only one of many examples where 1 would muck up various formulas and theorems if it was allowed to be prime. Once you see a dozen cases of this, it becomes really clear that mathematicians did the right thing by excluding 1 from being prime.)

This also explains why negative integers are not considered prime. Example:

6 = 2 * 3
6 = -2 * -3

Since "negative primes" would also destroy the theorem, they are also excluded from being prime.

oops, I mean "positive integers", not "positive numbers" -- sorry.

Nobody said anything about considering 1 a prime. It still isn't a prime in OP's definition.