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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"?

Blue
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Shay Brokensha
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  • I dont know exactly but isnt it natural to restrict the domain to the naturals? – Hermis14 Dec 21 '21 at 01:28
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    Related: https://math.stackexchange.com/questions/4332152/unique-factorisation-theorem-for-mathbbz-setminus-0/4332164#4332164 – Ethan Bolker Dec 21 '21 at 01:28
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    We really want Unique Factorization to be true. $6=2\times 3=(-2)\times (-3)$ is not meant to be two distinct factorings. – lulu Dec 21 '21 at 01:37
  • No. The presumed domain of discourse is the natural numbers. If we talked about multiplying -1 and -7, we'd go beyond that domain. Also, such a prime number is equal to the product of many pairs of numbers, like 14 and .5, 28 and .25, 42 and .125, and as the pattern I would make clear an infinity of pairs of numbers. – Doug Spoonwood Dec 21 '21 at 01:37
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    You could do that, but would then want to consider $-7$ as a prime, and to think of $7$ and $-7$ as being the same in some sense when you think about factoring. What's important is to get to the assertion that an integer is a product of primes in essentially one way. Related: https://math.stackexchange.com/questions/4332152/unique-factorisation-theorem-for-mathbbz-setminus-0/4332164#4332164 . – Ethan Bolker Dec 21 '21 at 01:38
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    "a number that is divisible only by itself and 1" is something of a bumper-sticker definition: short and easy to remember, but lacking in nuance. So, your concern about it is warranted. However, in proper context, it is understood that the divisors under consideration are positive integers. (The "integer" part is important, too, or else one could raise concerns about the fact that, say, $7 = 2\cdot\frac{7}{2}$.) – Blue Dec 21 '21 at 01:38

2 Answers2

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This is a good question, and I think it motivates the more general definition of a prime number. Let's begin with units.

A unit is an integer such that it has an integer inverse.

In other words, a unit is an integer $k$ such that there is an integer $l$, with $k\cdot l=1$. What are the units? Well, it is obvious that $1$ is a unit, but $-1$ is also a unit, since $-1\cdot-1=1$. However, there are no other units (can you prove why)?

Now, to prime numbers:

An integer $k$ is prime if it is not a unit, and for any integers $a,b$ such that $a\cdot b=k$, either $a$ or $b$ is a unit.

In other words, a prime number is one that can't be a product of two non-unit integers.

So what are the primes? Well, all of the primes you know ($2,3,5...$) are primes, but so are $-2,-3,-5,...$

The wonderful part about these definitions is that they generalize nicely to more interesting sets of numbers.

P.S. If you learn abstract algebra, you'll see that I've kinda lied to you. The definition of prime numbers I've given you is actually that of irreducible numbers, but for the integers, they are one and the same.

Rushabh Mehta
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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.

Tiago Cavalcante
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