# Primes, composites, and units

You may know that the prime numbers are 2, 3, 5, 7, 11, 13, 17, etc.; you may even know that a prime number is a natural numbers which:

• doesn't have divisors other than 1 and itself,
• have only two divisors,
• cannot be factorized further,

etc. etc. These descriptions conveys the idea, but are not precise enough. For example, some people get pedantic when interpreting the word ``and'' in ``1 and itself''. Also, you are left wondering why mathematicians and books exclude 1 as a prime.

The school system did a confusing job in describing primes and composites, leaving more questions than answers.

## Units

Definition: a unit is a number which has a reciprocal (inverse, multiplicative inverse); x is a unit if and only if for some y, x*y = 1.

(Normally, you would write 1/x or x^(-1) for y.)

Now whether or not a number has a reciprocal depends on which number system you work with. If you confine yourself to the integers, 2 does not have a reciprocal, because nothing multiplies 2 to give you 1. But if you allow fractions, then 0.5 or 1/2 is the reciprocal of 2.

In number theory, we usually confine ourselves to the natural numbers, sometimes to the integers; fractions play an auxiliary role only. Thus, in the natural numbers, 1 is the only unit; in the integers, 1 and -1 are the only units.

Related to the concept of units is the concept of associates.

Definition: v and w are associates if and only if you can multiply a unit to one and get the other, i.e., for some unit u, u*v = w. Note that when this holds, you can also find some unit t so that v = t*w, namely, choose t to be 1/u since u is a unit.

So for example, 3 and -3 are associates; 3 and 3 are associates too. The concept of associates is not of much use other than stating a certain theorem later.

## Primes and composites

Definition: a prime is a number which is not 0, not a unit, and not the product of two non-units; x is a prime if and only if x != 0, x is not a unit, and if x = d*e then d or e is a unit.

The idea is that a prime number cannot be further ``factorized'' into a product of two factors; if you force it, one of the factors will become a unit like 1 or -1 which is rather insignificant, and consequently the other factor is an associate, which is ``not much different'' from the original number (like, differ by a sign only), so you are not really factorizing anything.

So for example, 2, 3, 5, 7, 11, 13, 17, 19 are primes, because there is no way you can factorize them other than writing something boring like 1*2, (-1)*(-3), 5*1, etc.

The following theorem is sometimes used as an alternative definition of primes.

Theorem: x is a prime if and only if x is not 0, not a unit, and if x | s*t then x | s or x | t.

In other words, when a prime divides a product of two numbers, it must also divide at least one of them. This can easily be understood from the factorization point of view: a prime factor of the product s*t must come from s or from t, otherwise how can it be a prime factor of s*t to begin with. Although this idea as stated here is not precise enough to constitute a proof, it is the spirit of the proof. Anyway, this theorem about primes turns out to be more important than the definition.

Definition: a composite is a number which is not 0, not a unit, and not a prime.

So for example, 6, 8, 9, 10, 12, 15 are composites. A charaterizing property of composites is that they can be factorized, like 6 = 2*3, 12 = 4*3 = 2*6 = 2*2*3, etc.