Understanding the Definition and Status of -1 as a Prime Number

[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.

 

[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..

 

[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".

 

[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.

 

[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)

 

[Icebreaker]

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

 

[arildno]

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

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

 

[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?

 

[arildno]

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

 

[robert Ihnot]

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

(-1)(-1) = 1.

 

[arildno]

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

(-1)(-1) = 1.

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)

 

[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.

 

[shmoe]

When we talk about "Prime Numbers" it's generally assumed you are referring 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.

 

[Hurkyl]

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)

 

[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.

 

[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"...

 

[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.

 

[matt grime]

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"...

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.

 

[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.

 

[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.

 

[mathwonk]

although this attitude may seem backwards, actually it is a very practical way to proceed: i.e. definitions should be given so that they make they naturally stated theorems true.

of course the theorem does not say that, but almost, it says the factorization is unique except for ordering and multiplication of prime factors by units.


i.e. 2*3 is equated with 3*2 and also with (-2)*(-3), but so what?

at least the number of factors is well determined when we exclude units.

 

[DaveC426913]

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)

I don't understand this definition.
If ab=60 because a=12 and b=5, does that mean p can equal 4 (because it divides a)? Clearly not. Where have I gone wrong?

 

[matt grime]

I don't understand this definition.
If ab=60 because a=12 and b=5, does that mean p can equal 4 (because it divides a)? Clearly not. Where have I gone wrong?

you aren'rt using quantifiers properly

4 divides 4=2.2 but does not divide either 2 or 2 so 4 is not prime.

p is prime if p is not a unit and if p|ab then p|a or p|b for all a and b

 

[xykomath]

Is -1 a prime number?

1 can be prime just like 2 or 3 since 1 is indivisible by any other natural number besides 1 and oneself, which is again 1, but we don't want 1 to be a prime number. The real reason everyone in the universe wouldn't want 1 to be a prime is that 1 would SPOIL a perfectly cool theorem, the fundamental theorem of arithmetic (FTA), not because 1 contradicts FTA. how so? before proving FTA we made a choice not to let 1 be a prime, so 1 can't come back contradict FTA, right? ok, why did we choose not to choose 1 to be a prime? let's recall what FTA say; every positive integer greater than 1 can be expessed UNIQUELY as a product of primes up to the ordering. Remember, like all the book said, primes are the number 2,3,5,7,11,...(2^25964951)-1,...
Thus IFFFFFFFFFF 1 were a prime then multiplying 1 to the product of primes doesn't change the value/magnitude of the product but only put more factors into an already nice looking product of primes,i.e. 1 would mess up an important property an interesting claim, the UNIQUENESS in the FTA. now you see that our choice not to let 1 be a prime was a right choice? it's important to make a good choice and a right one too, right? try this with an example, say a PERFECTly good positive integer greater than 1, 6. FTA says; 6=2x3, a unique product of primes 2 and 3 up to the ordering (3x2).
Look what good can come of 1 being a prime? 6=1x2x3=1x1x2x3=1x1x1x1x1x1x1x1x1x1xxxxxxxxxxxxxxxxxxxxx1x2x3,
nothing . 1 just going to make a nice looking product of primes, 2x3, look bad. therefore we ought to be happy with just 2,3,5,7,11,... being primes . there is enough prime numbers for everyONE.

 

[brunardot]

The determination as to what numbers are prime numbers depends on the definition of prime numbers.

The conventional definition of prime numbers is contrived as it is the result of the combination of several sets that can be considered as unrelated, which creates the problems of analysis that exists.

I consider that:

Mathematics does not
explain Nature;
Nature explains
mathematics.

All mathematics is a function of Nature;
thus, its sublime poetry . . .

Thus, minus one cannot be a prime as Nature does not recognize negative numbers, as there is no "up" or "down" in the Cosmos. The closest that Nature comes to such recognition is the relative difference between the crest and trough of a wave.

Natural prime numbers are uniform in their distribution; that is, they can be mapped with a simple algebraic function, to the sequence of Natural integers: 0, 1, 2, 3, 4, 5...

Natural prime numbers are generated by the hypotenuse of any ellipse, relative to the integer value of the perigee.

 

[Hurkyl]

Thankfully, mathematicians do their work based on definitions, not poetry.

Mathematicians make definitions because the things they define are useful. And yes, sometimes we choose definitions precisely because they make certain theorems easy to state.

For example, take the fundamental theorem of algebra: evey polynomial of degree n has exactly n roots.

This is, of course, only true with the appropriate definition of root (i.e. can be complex), and when using the appropriate method of counting them (i.e. counting multiplicities).


As for the definition of prime, it is more useful for it to exclude units (e.g. 1 and -1, when living in the integers) than to include units.