I’ve never been sure of this. Some people think it is, some think it isn’t. What’s the Straight Dope?

Prime numbers must have two factors. The number one has only one factor, itself.

From http://mathforum.org/dr.math/problems/christy10.6.97.html

There you go: not prime and not composite.

I have nothing further to add.

[fixed coding]

[Edited by bibliophage on 10-18-2001 at 04:52 AM]

Thanks guys. That was bugging me.

cant be. all prime numbers fit this formula: 2 to the nth power + 1 . not that doesnt mean that ALL numbers that fit that are prime, but say 17 is prime because 17 = 2 rasied to the 4th + 1. you cant get the number one to fit this.

Can’t be true; 7 is prime. However, if a prime is of the form 2[sup]n[/sup]+1, I believe you can derive a perfect number from it.

FWIW, one is what’s called a unit. In any system which has arithmetic that follows the same rules as the real numbers, there are four types of numbers: primes, composites, units, and zero divisors. These categories are (IIRC) mutually exclusive.

I assume everyone’s right and one is not prime, but now I’m curious as to why the definition for a prime number seems have been made to explicitly eliminate one as a prime number. Take Seraphim’s definition: “A prime number is one with exactly two positive divisors, itself and one. One has only one positive divisor.”

I’m reminded of a thread a few months back that discussed the meaning of any number taken to the power of zero. IIRC that was defined as always being equal to 1 even though there was no fundamental reason for it. It was more of a convention that was accepted to make other calcualtions work.

So I guess I’m asking if there something behind defining “prime” so as to exclude one.

Yes-if one is prime, then the Fundamental Theorem of Arithmetic is false. As you can guess from the name, this would suck. The FTA states that there is exactly one prime factorization for any composite number. Eg. 12 can be factored as 2*2*3, and there is no other collection of primes that yields 12 when multiplied together. If 1 is considered prime, then this isn’t true-for instance, there’s 1*2*2*3, 1*1*2*2*3, etc. Now, you could re-define the FTA to account for this-but it makes a lot of things a lot easier to just leave 1 out of this altogether …

A non-unit p is prime iff if p divides ab then p divides a or p divides b or p divides both a and b.

by this dfiniton -p would also be a prime. Ok.I heard this long ago…they are called associative primes(i think…)

**Googler**: Not sure what you’re talking about here. If p=6, a=12, b=18 your criterion is met but 6 is non-prime.

Theorem: If a prime p divides ab, then p divides a or p divides b.

Note that the mathematical “or” is non-exclusive, so this allows that p divides a and p divides b.

The statement “if p divides ab then p divides a or p divides b” has to be true for all a and b for p to qualify as a prime. So for p=6, while a=12, b=18 does work the combination a=4, b=9 does not.

**NevarMore**: “all prime numbers fit this formula: 2 to the nth power + 1 .”

What you’re thinking of are called Mersenne primes, and they fit the form 2[sup]n[/sup] - 1. The smallest such examples are 3 (n = 2), 7 (n = 3), and 31 (n = 5). Each Mersenne prime corresponds to an even perfect number, and vice versa. If 2[sup]n[/sup] - 1 is prime, then (2[sup]n - 1[/sup])(2[sup]n[/sup] - 1) is perfect. The smallest perfect numbers are 6 (n = 2), 28 (n = 3), and 496 (n = 5). There are, last I heard, 60-something Mersenne primes known, and it is not known whether there are an infinite number or not. Conversely, it is known that there are infinite number of primes.

If 1 were prime, then 2[sup]n[/sup] - 1 would be prime for n = 1, and so 1 would be perfect. But, as has been said, it’s not.

Actually, what you’re describing is a subclass of primes called Germain primes, which are all of the form 2[sup]N[/sup] + 1. As others have pointed out, not all primes can be written as that.

What you **meant** to say was that all prime numbers can be written as either 4n + 1 or 4n - 1. This is called Fermat’s Prime Theorem (not to be confused with his Last Theorem). Predictably, he left it for others to prove this theorem too.

**Seraphim**

Actually, Germain primes are something else, click the link for more info.

Maybe **NevarMore** was thinking of Fermat primes–primes of the form 2[sup]2[sup]n[/sup][/sup] + 1. (Not all primes are of this form, of course, and it’s not known how many Fermat primes there are).

Finally:

I find this impossible to believe. Number one, it’s not true (2 is a prime not of this form). Number two, if we make the obvious correction (All primes other than 2 can be written as either 4n+1 or 4n-1), then this simply states the obvious fact that all primes (other than 2) are odd. I’m sure this was common knowledge long before Fermat.

An integral prime is a complex integral prime if and only if it is of the form 4n - 1.

2 is not of the form and can be written as (1+*i*)(1-*i*).

3 is 2*2 - 1 and is a complex prime.

5 is not of the form and can be written (4+*i*)(4-*i*).

That is the only thing I can think of about whether a prime can be written as 4n - 1 or not. Is that what you were thinking of?

In answer to **bnorton**:

One reason I think it’s important to consider 1 not to be a prime number is it kind of messes up the way composite numbers are considered. A composite number can be broken down into a *limited* number of factors (all of which have to integers). You can break those factors that are down into composites into their factors, and so on, until all you’ve got is a bunch (but again, a limited bunch) of prime numbers.

If you let one into the mix, the lid blows off. The number 12, when broken down into primes strictly, is just 2x2x3. If you consider one a prime, it could be 1x1x2x1x1x2x3 and you’d never really be done.

The point is, one just isn’t important as a factor; you can multiply a number by one a kazillion times and it wouldn’t change. Only the true primes are important as factors.

I italicized most of my post and I totally repeated what SCSimmons had to say. Redundant and poorly formatted.

Dang, I thought the format looked familiar, but I should have done some more research first. Mea culpa.

Well, I condensed things. What Fermat said in full was that primes of the form of 4n+1 are always the sum of two squares, whereas those of the form 4n-1 can never be written this way. This site elaborates more on that theorem. I still stand by my assertion that all primes (other than 2) can be expressed as either of these forms.