Rrerku

I'm having a bit of trouble understanding why this incorrect definition of primality only defines one prime number.

"p is prime if it is divisible by only itself and 1."

My understood definition of a prime number is a positive integer that has exactly 2 positive integer factors: 1 and itself. Is the problem with the definition related to the fact that it doesn't specify sign, so then -1 would be the only "prime" number?

Sorry if my question is related to others, I am quite new to number theory and could not find any other resources that would help me with this specific problem.

Usually, in beginning number theory, we work only in the natural numbers (usually including $0$) so there are no negative numbers in this context. Assuming that, the definition of "divisor" would include the fact that a divisor is a natural number, than therefore positive. Then the definition:

A (natural) number $p$ greater than $1$ is prime iff it is divisible only by itself and $1$.

The "greater than $1$" part is important, because we don't want $1$ to be prime. (Although, historically it was considered a prime.) The implicit positiveness of the potential divisors is also important.

Later in the course, you can extend these definitions of "prime" and "divisor" to include the negative integers and then you'll have to adjust your definitions a bit.

Is the problem with the definition related to the fact that it doesn't specify sign, so then -1 would be the only "prime" number?

Actually is worse. There would be no prime numbers as for any $x in mathbb R$ then $x = n*frac xn$ for any integer $n$ so $x$ has infinitely many divisors and can't be prime. $3$ is not prime because $frac 32, frac 34, frac 35$, etc all divide $3$. (The definition of $a$ divides $b$ is that there exists an integer $k$ so that $b = ka$. There is nothing in the definition that says $a$ and $b$ are integers)

So.......

The definition of "Having only $1$ and itself as divisor" assumes that we are only considering natural numbers (i.e. positive integers) and even then we are considering $1$ and "itself" to be different.

This definition is not even considering negative numbers to be considered. Nor is it considering that that rational non integers can be divisors.

Which is okay. We just have to be specific. The following would be a perfectly acceptable definition:

A natural number is prime if it has exactly two natural divisors; itself and $1$.

We can extend it extend this to all integers by saying a prime in any integer other than $pm 1$ with only two positive integer divisors; $1$ and the absolute value of itself. ... or a prime number is an integer which can not be expressed as a product of two or more non-unit ($pm 1$) integers.

With more abstract algebraic concepts we can define primes more precisely as Flowers' answer and various comment links explain. These also allow for primes in systems other than just the integers.

Your definition is correct. The confusion is stemming is from the fact that fundamental results and definitions-- both in courses and in mathematics in general-- are often times stated without certain underlying hypotheses that are understood "automatically," whether for technically sound reasons or nothing better than "this is number theory so this is how we do it." I will address each of your statements.

"p is prime if it is divisible by only itself and 1."

This is the highly traditional definition of prime that is usually provided in number theory courses. This definition comes with an unstated condition that only positive divisors are considered. This is done to simplify and streamline many of the results in elementary number theory. (Notice in a similar vein how prime numbers are also usually only defined over the positive naturals. We never really hear about "negative primes," and we generally state that "2 is the only even prime," despite the fact that -2 also satisfies the same conditions.)

My understood definition of a prime number is a positive integer that has exactly 2 positive integer factors: 1 and itself.

Your definition is correct. In fact, your definition is actually more precise, and arguably "better," at least in the sense that it does not require the reader to "import" the assumption that only positive divisors are being considered in context.

Is the problem with the definition related to the fact that it doesn't specify sign, so then -1 would be the only "prime" number?

Taken literally, yes, the first "traditional" definition of prime implies this statement, but of course as I said the traditional definition is importing the positive divisor assumption. So your statement is a humorous but untrue technicality-- we know -1 is not the only prime number. As I also mentioned earlier, we usually don't even allow primes to be negative.

The precise way to define prime and irreducible elements in a commutative ring are given here and here. For the integers, the definitions are equivalent, so you can use either. Rephrasing the second link for integers:

A unit in a ring is an element that has a multiplicative inverse. In the integers, the only units are $1$ and $-1$. So a nonzero non-unit element of the integers is irreducible (prime) if it cannot be written as the product of two integers which are both non-units (not 1 or -1).

Note that this implies that negative primes exist. Unique factorization is defined up to multiplication by units.