Is one considered a prime number?

[mateomy]

Is one considered a prime number? I know the definition of a prime number is any number that is constituted only by 1 and itself; does this include one? Why or why not?

 

[mathman]

It is mainly a matter of convenience not to call 1 a prime. Otherwise many theorems involving prime numbers would have to say "excluding 1". For example every integer can be expressed uniquely as a product of powers of primes. If you included 1, you would have to say "except 1" to have uniqueness.

 

[micromass]

No, 1 is not a prime number. This is quite an arbitrary definition, but there are good reasons not to define 1 a prime number.

For example, every number has a unique prime factorization, eg

60=2².3.5

But if 1 is a prime, then this is not true anymore, because

60=2².3.5.1

is then another prime factorization.

There are another quite nice results for which 1 should not be prime. So 1 is not prime. A prime number is in fact, by definition, a number with exactly two divisors. This definition would exclude 1.

 

[mateomy]

Oh okay, that makes sense. Does anyone have any reading suggestions on primes specifically. I just started reading "Elementary Theory of Numbers" by LeVeque. Any critiques on this, or pointers to another text perhaps?

Thanks.