Definition:Prime Number

Definition
A prime number $$p$$ is a positive integer that has exactly two positive divisors.

Those two divisors of $$p$$ are $$1$$ and $$p$$, from Integer Divisor Results:


 * $$1$$ divides all the integers;
 * Every integer divides itself.

Equivalent Definition
$$p$$ is prime iff $$\tau \left({p}\right) = 2$$, where $$\tau \left({p}\right)$$ is the tau function of $$p$$.

Odd Prime
Every even integer is divisible by $$2$$, which is what "even" means. Therefore, apart from $$2$$ itself, all primes are odd.

So, referring to an odd prime is a convenient way of specifying that a number is prime, but not equal to $$2$$.

Composite
An integer greater than $$1$$ which is not prime is defined as composite.

Comment
It follows from this that $$1$$ is not a prime number by this definition, as $$1$$ has only one positive integral factor, that is, $$1$$ itself.

The wording of this definition saves having to make a special case for $$1$$, which (for all sorts of reasons) is not considered to be a prime number.

Some authors use the symbol $$\mathbb{P}$$ to denote the set of all primes. This notation is not standard (but perhaps it ought to be).

The letter $$p$$ is often used to denote a general element of $$\mathbb{P}$$, the same $$n$$ is often used to denote a general element of $$\N$$.