Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In number theory, is prime number usually defined to be some natural number or some integer, i.e., must it be positive or can it be either positive or negative?

Thanks and regards!

share|improve this question
    

2 Answers 2

up vote 8 down vote accepted

Today, the definition depends on context. But the definition that applies to arbitrary rings is usually along the lines of:

A nonzero element $p$ of a ring $R$ is a prime if and only if $p$ is not a unit, and whenever $p|ab$, either $p|a$ or $p|b$.

(Sometimes one also allows $0$, which is a prime number in $R$ if and only if $R$ has no "zero divisors". In the integers, allowing it has virtues and minuses, usually reflected in exclusion clauses in theorems depending on whether you admit it or not).

If we take this definition and apply it to $\mathbb{Z}$, then negative numbers such as $-2$, $-7$, etc., are "primes." This leads to some difficulties with the Fundamental Theorem of Arithmetic (and its generalization to UFDs), in that factorization into primes is no longer unique: $4 = 2\times 2 = (-2)\times(-2)$.

But divisibility also defines an equivalence relation, "are associates": we say that $a$ and $b$ are associates if and only if $a|b$ and $b|a$, which for rings like $\mathbb{Z}$ that are domains (no zero divisors, and a multiplicative identity) is equivalent to saying that there is a unit $u$ such that $a=ub$. So we associate prime elements with each other if they differ by a unit, so that we consider both $2$ and $-2$ to be "essentially the same" for divisibility purposes. The fundamental theorem of arithmetic then says that the factorization is unique "up to order and associates" (so you may replace some primes by other primes that are associates of them, such as replacing $2$ by $-2$).

These days in number theory.... if you are leading towards algebraic number theory, the definition of prime will likely encompass both the positive and negative versions, because it facilitates the passage to things like primes in $\mathbb{Z}[i]$, and the consideration of prime ideals later on. But in very elementary, beginning, number theory when you are dealing only with natural numbers, you'll likely stick to the natural number definition.

Because in any ring $a|b$ if and only if for any units $u$ and $v$ you have $ua|vb$, and because the property of being a prime number has to do with divisibility, it usually doesn't matter whether you are restrictive or not (provided that, if you allow negative numbers as primes, you add an 'up to associates' clause whenever needed).

share|improve this answer

These 3 answers pretty much cover everything. I agree with all of them, but mostly with #3.

share|improve this answer
    
For example, (per answer #1) Euclid's Elements has lots of information about primes, but nothing at all about negative numbers. –  GEdgar May 7 '11 at 19:58
    
Alas, the linked Prime Pages answer does not address one of the most important conventions: why/when 0 is prime? –  Bill Dubuque May 7 '11 at 20:14

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.