Prime numbers??

Nappy

Why is'nt 1 a prime?

A prime number is one that is only divisible by 1 and itself...
So why is 1 going under the radar or am I missing something??

seanybiker

with 1 being the reason for prime numbers its just let go. I dont know the mathimatical reasoning but thats it in my terms lol.

Col Man

I think it's something to do with prime factorisation. Every positive number (except 1...lol) can be represented as a unique product of primes (that is, prime numbers multiplied together). But if 1 is counted as prime, this no longer holds true, as 330 goes from 2 x 3 x 5 x 11 (the only way to represent it as product of primes) to 1x2x3x5x11, or 1x1x1x1x1x2x3x5x11.

Might not be the reason at all....

Brethitmanhart

The exact definition of a prime number "a natural number that has exactly two distinct natural number divisors." For all the other numbers the two natural numbers that will have to divide into it are 1 and themselves. But 1 isn't a prime number because of that proper definition above...it only has 1 natural number divisor.

Brethitmanhart

Col Man wrote: »

I think it's something to do with prime factorisation. Every positive number (except 1...lol) can be represented as a unique product of primes (that is, prime numbers multiplied together). But if 1 is counted as prime, this no longer holds true, as 330 goes from 2 x 3 x 5 x 11 (the only way to represent it as product of primes) to 1x2x3x5x11, or 1x1x1x1x1x2x3x5x11.

Might not be the reason at all....

Your reasoning is the reason why it's important that 1 isn't counted as a prime number.

RoundTower

Col Man wrote: »

I think it's something to do with prime factorisation. Every positive number (except 1...lol) can be represented as a unique product of primes (that is, prime numbers multiplied together). But if 1 is counted as prime, this no longer holds true, as 330 goes from 2 x 3 x 5 x 11 (the only way to represent it as product of primes) to 1x2x3x5x11, or 1x1x1x1x1x2x3x5x11.

Might not be the reason at all....

you can think of it that 1 can be represented as a unique product of primes too, to make all the definitions more complete. It's what you get when you multiply no primes together.

Fremen

As Brethitmanhart said, if 1 is considered to be a prime, then you lose unique factorisation. Other theorems and definitions also become messier. A couple of examples:

Let p be a prime, and suppose 1 is a prime. We have
p = 1.p
so p is a product of two primes, so p is composite. This is clearly a contradiction, so we need to revise our definition of composite numbers.

Sqrt(p) is not always irrational if 1 is a prime.

Col Man

Fremen wrote: »

As Brethitmanhart said, if 1 is considered to be a prime, then you lose unique factorisation.

You mean... as Col Man said...? Ha sorry

Julia Set

Actually, the very definition of a prime number specifically excludes 1: An integer n is called prime if n > 1 and if the only positive divisors of n are 1 and n.

(For those interested, this definition is taken from T. M. Apostol's Introduction to Analytic Number Theory, Springer-Verlag Series Undergraduate Texts in Mathematics.)

Col Man

Lol it's a bit unfair to say that, I mean n>1 is only a rather recent (in relative terms) addition. I think OP knows that 1 isn't prime, he's asking why. Saying n>1 is basically saying "cos is isn't prime", which isn't all to helpful for OP

Fremen

Col Man wrote: »

You mean... as Col Man said...? Ha sorry

Yes, sorry, my Brainwrong.

Julia Set wrote: »

Actually, the very definition of a prime number specifically excludes 1: An integer n is called prime if n > 1 and if the only positive divisors of n are 1 and n.

(For those interested, this definition is taken from T. M. Apostol's Introduction to Analytic Number Theory, Springer-Verlag Series Undergraduate Texts in Mathematics.)

Yep, and Apostol nicked it from Hardy and Wright, so I guess it's been around in that form since at least 1938

Julia Set

Sorry, Col Man. I realise that now, and I was just going to edit my last post, but you beat me to it!

Returning to the topic, though, the number 1 seems to occupy a special position in that it is considered neither prime nor composite (i.e. non-prime). You've already demonstrated good reasons for leaving 1 aside, so I'll just concur with that.

Capt'n Midnight

when we meet the aliens we can ask them if they consider 1 to be prime

interestingly if they use an odd base like 9 then they wouldn't consider 2 to be the only even prime

MathsManiac

Capt'n Midnight wrote: »

interestingly if they use an odd base like 9 then they wouldn't consider 2 to be the only even prime

Why do you say that? Neither the definition of "prime" nor the definition of "even" is dependent on the base. So 2 is the only even prime, no matter what base you work in.

timbrophy

The definitions of a prime number originate in ancient Greece. The Greeks regarded 1 as the generator of all numbers rather than a number itself.

Tim

Tau

I think it's simpler than all that. If you look at the first few proofs in any basic number theory course, you quickly realise that you've got two choices:

1) Say that 1 isn't prime

or 2) go around all the time saying "all the primes except one" and "p is prime but p is not 1" and so on.

Saying that 1 isn't prime is just easier.

I could paraphrase what I'm saying by saying that it's quite useful to be able to refer to the set {2,3,5,7,11, ... } with one word, but you almost never need to be able to refer to the set {1,2,3,5,7,11, ...}