Shouldn't the definition of a prime number be changed to account for negative factors? At the moment, from what I can gather the current definition of a Prime Number is; "a number that is divisible only by itself and $1$ (e.g. $2, 3, 5, 7, 11$)". However such a prime number like $7$ can also be made by multiplying $-1$ and $-7$. Hence shouldn't the definition be changed to "it can only be divisible by itself and one as well as $-1$ and its negative counterpart"?

Wikipedia says that a prime is not a product of other natural numbers:

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

