# Is -1 a prime number?

1. Jun 8, 2005

### AntonVrba

We all know the definition of prime numbers and the first prime number is always 2.

Why is -1 not listed as a prime number? , it qualifies as it passes all tests for a prime number.

2. Jun 8, 2005

### arildno

Well, I've only seen prime numbers defined on N.
I'm sure some mathematician has generalized the concept properly, but I can't help you on that..

3. Jun 8, 2005

### HallsofIvy

Well, I know the definition of "prime number" but apparently you don't. Every definition I have seen starts "an integer greater than 1 such that... " or "a positive integer such that ..." immediately excluding -1 since -1 does not "pass all the tests".

4. Jun 8, 2005

### dextercioby

Didn't you find surprising/suspicious that,when being taught in school the algorithm of Eratosthenes,you didn't include the negative integers...?

Daniel.

5. Jun 8, 2005

### matt grime

One *may* define primes on the integers (or any other ring) but -1 still fails to be prime even after you have defined this extension to the integers (it is a unit, that is a divisor of 1)

6. Jun 8, 2005

### Icebreaker

But would defining primes on integers not contradict the fundamental theorem of arithmetic?

7. Jun 8, 2005

### arildno

Wouldn't it rather be the other way around..
(I'm sure a modified version of the FOTA will hold, though..)

8. Jun 8, 2005

### Icebreaker

Mmm, I haven't thought of contradictions as a one-way street... Then again I'm not entirely familiar with these subtle rules of mathematical logic :(

I guess of one modifies fota as 'absolute values' it will hold; but that seems like cheating, doesn't it?

9. Jun 8, 2005

### arildno

What I meant, is that whatever definitions and axioms you make is prior to any theorem you might derive from them..

10. Jun 8, 2005

### robert Ihnot

Icebreaker has an interesting point: But would defining primes on integers not contradict the fundamental theorem of arithmetic?

(-1)(-1) = 1.

11. Jun 8, 2005

### arildno

Which shows that if you extend your definition of "primes"& "factorization" in the simplest manner, then you cannot derive FOTA as a valid theorem with this new prime set.

(Besides, I'd rather use (-2)*(-3)=2*3, or something like that)

Last edited: Jun 8, 2005
12. Jun 9, 2005

### marteinson

I think -1 should be prime, or should not be prime, for the same reasons 1 itself is or is not considered prime. For the record, I don't even think 2 and 3 should be considered the primes in the same way 5, 7, 11, 13 and the rest. In a way, 1, 2 and 3 are too small to be divisible by anything other than 1 and themselves, which is different to larger numbers being structurally composed of such a number of elements that they are indivisible, like 5039.

13. Jun 9, 2005

### shmoe

When we talk about "Prime Numbers" it's generally assumed you are refering to the naturals. You can talk about primes or irreducibles in the integrers, but honestly once you know 6=(2)(3) it's not terribly exciting to write 6=(-2)(-3), so we generally restrict ourselves to the naturals, because the integers are essentially the same.

Anyways, the "proper" generalization would be to prime ideals in the integers. I call this "proper" because unique factorization will hold in this and more general settings where we consider the integers in different number fields. One complication that's removed by this generalization is the annoyance of units. If you call ..5, 3, 2, -2, -3, -5, ... primes in the integers, you'd have unique factorization up to multiplication by the non-trivial unit -1 sprinkled in. If you're looking at ideals, the ideal generated by 2 and the ideal generated by -2 is the same thing. This might not seem like a big deal, but in other instances you'll have more units kicking about to muck things up.

14. Jun 9, 2005

### Hurkyl

Staff Emeritus
One of the most important properties of primes is that if p is prime, then whenever p divides a b, then p divides a or p divides b. (or both)

In fact, in the general case, this property is taken to be the definition of a prime -- a number for which that property is true.

In general, this is different than irreducibility. n is irreducible iff, for any factor a of n, a must either be a unit, or equal to n times a unit.

(A unit is something invertible)

15. Jun 9, 2005

### shmoe

I should have been more careful- I didn't mean to give the impression that "prime" and "irreducible" were in general the same concept (I'm not even sure why I bothered complicating things by mentioning irreducible). Thank you for adding the clarification.

16. Jun 9, 2005

### Icebreaker

Primes have the property of having only the trivial divisors 1 and itself. However, if we allow negative integers to be defined as primes, that will no longer be the case. Then again if we speak in "absolute values"...

17. Jun 10, 2005

### matt grime

The fundamental theorem is defined on N. What has that got to do with the integers? There is a different version of the FOTA that states in Z every number is the product of primes in an essentially unique way, where essentially unique means that up to multipliying by units (+/-1) things are the same. One can define primes for any ring and most of them will not have unique factorization into primes.

18. Jun 10, 2005

### matt grime

that may be your definition, and a useful ad hoc one it is too, but that isn't the formal definition for an arbitrary ring which is what the integers are.

19. Jun 11, 2005

### mathwonk

as matt tried to explain, a prime number can never be a "unit", i.e. a number with a multiplicative inverse.

in an arbitrary unique factorization domain, one tries to determine:

1) the units,

2) the primes.

obviously -1 is a unit, since (-1)(-1) = 1. hence it is never a prime.

the correct definition of prime integer is: any integer n which is not a unit, and such that if n = ab, with both a,b integers, then one of a or b must be a unit. thus -1 is not a prime, but -5 is.

another definition, if you know some algebra, is that an integer is prime if and only if the ideal it generates is a (proper) prime ideal.

20. Jun 12, 2005

### murshid_islam

why 1 is not a prime number

the fundamental theorem of arithmetic is that every number can be uniquely factorized into prime numbers. e.g., 6 can be factorized as 2*3. but if 1 is aprime number then this theorem doesn't hold true. then 6 cannot be uniquely factorized into primes. so 6 = 2*3 or 6 = 1*2*3.