# Why is 1 not considered a prime number?

HallsofIvy,

Just a thought but your reasoning is slightly off

" Calling 1 a prime would make it untrue since then we could write 6= 1*2*3 or 6= 12*2*3 or..."

By the inclusion of 1 - you do not change the numebr of ways a number can be expressed as 1^x is always reducable back to 1 as long as x is a +ve integer (x must be a positive integer otherwise the 1^x term would not be a prime).

So actually 1 - by use of the fundamental theory is not excluded from being prime.

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NewScientist said:
HallsofIvy,

Just a thought but your reasoning is slightly off

" Calling 1 a prime would make it untrue since then we could write 6= 1*2*3 or 6= 12*2*3 or..."

By the inclusion of 1 - you do not change the numebr of ways a number can be expressed as 1^x is always reducable back to 1 as long as x is a +ve integer (x must be a positive integer otherwise the 1^x term would not be a prime).

So actually 1 - by use of the fundamental theory is not excluded from being prime.

If 1 is a prime, then 3*2*1 and 3*2 and 3*2*1*1 are three different ways of factoring 6 into primes. Pointing out that they all can be reduced to a canonical factorization by eliminating the extra ones doesn't change that fact. You still have different factorizations.

"x must be a positive integer otherwise the 1^x term would not be a prime"? That doesn't make sense; 1^x = 1 for any number x, so if 1 is a prime number than so is 1^x, for any x.

master_coda said:
If 1 is a prime, then 3*2*1 and 3*2 and 3*2*1*1 are three different ways of factoring 6 into primes. Pointing out that they all can be reduced to a canonical factorization by eliminating the extra ones doesn't change that fact. You still have different factorizations.

Well..."every positive integer can be written as a product of powers of primes in exactly one way." To express 3, a sa product of primes (it is a positive integer and so must be expressable as a product of primes), one must use 1. if one is not prime (as I know it is not) it does not omit it form this function does it?!

master_coda said:
"x must be a positive integer otherwise the 1^x term would not be a prime"? That doesn't make sense; 1^x = 1 for any number x, so if 1 is a prime number than so is 1^x, for any x.

sorry - I have been dealing with a y^x problem and hadn't disassociated myself from it!

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AKG
Homework Helper
EL said:
How about the positive integer "1" then?
Let pi be the ith prime (so p1 = 2, p2 = 3, etc.). Then

$$1 = \prod _{i = 1} ^{\infty } p_i^0$$

This is unique since 1 is the only integer for which all the exponents of the primes are zeroes.

matt grime
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NewScientist said:
Well..."every positive integer can be written as a product of powers of primes in exactly one way." To express 3, a sa product of primes (it is a positive integer and so must be expressable as a product of primes), one must use 1. if one is not prime (as I know it is not) it does not omit it form this function does it?!!

no, one mustn't. a single number on its own is a product.

EL
AKG said:
Let pi be the ith prime (so p1 = 2, p2 = 3, etc.). Then

$$1 = \prod _{i = 1} ^{\infty } p_i^0$$

This is unique since 1 is the only integer for which all the exponents of the primes are zeroes.

Yeah, it was the uniqueness which disturbed me, but it's pretty obvious...

HallsofIvy said:
Do you consider the Fundamental Theorem of Arithmetic:
"Every positive integer can be written as a product of powers of primes in exactly one way"
stupid?

Calling 1 a prime would make it untrue since then we could write 6= 1*2*3 or 6= 12*2*3 or...
I don't see how it makes the theorem untrue... if you need a certain definition of primes for some theorem to work, which I don't see why in this case, then change the rule to make an exception for number 1.
I think it is more important to see the number 1 for what it is rather then to hold on to some correctable definition of positive integers or whatever.

Please tell me, why are primes interesting?
They are interesting because they are devisable by themself and 1 only and I see no reason why you should exclude 1 that definition. The official definition may say something else but until I find a good reason to exclude 1, I will keep it in my list of primes.

Icebreaker
Primes are interesting precisely because 1 isn't a prime. 1 is a divisor of every number, therefore defining 1 as a prime dulls the main purpose of primes.

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shmoe
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WeeDie said:
I don't see how it makes the theorem untrue... if you need a certain definition of primes for some theorem to work, which I don't see why in this case, then change the rule to make an exception for number 1.

Absolutely. Changing the definition of "prime" makes the fundamental theorem of arithmetic as it's currently worded false, but it doesn't change the math behind it- a rewording will make things correct but uglier to state. If you change the definition of prime to include 1 then you have hundreds upon hundreds of theorems to go and change, many of them it will be enough to change "let p be a prime" to "let p be a prime greater than 1". This in itself is a comelling reason to leave it as is and evidence that 1 behaves unlike a prime. Go through a number theory text and see how many theorems that begin "let p be a prime" and see if they hold if you allowed p to be 1. Each case where it fails is more evidence that 1 behaves differently from a prime and it makes sense to exclude it.

WeeDie said:
I think it is more important to see the number 1 for what it is rather then to hold on to some correctable definition of positive integers or whatever.

See 1 for what it is: the multilplicative inverse. It's on it's own. The fact that it divides everything makes it unlike any prime or composite and why the natruals are usual divided into 3 sets, primes, composites, and 1.

WeeDie said:
Please tell me, why are primes interesting?
They are interesting because they are devisable by themself and 1 only and I see no reason why you should exclude 1 that definition. The official definition may say something else but until I find a good reason to exclude 1, I will keep it in my list of primes.

You can think of primes as the building blocks of the integers (the fundamental theorem says as much). They have uses all over, the money topic these days would be cryptography.

You're free to keep 1 on your list of primes. You're free to define anything in math any way you like. I know that I tend to ignore people who choose to ignore widely adopted conventions. We have conventions for a reason, to make communication easier and if someone can't be bothered to follow even basic ones it's not really worth the effort to see what else they may have decided to change.

arildno
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Gold Member
Dearly Missed
In my opinion, there is enough ugliness in the formulation of theorems concerning primes due to the pesky 2. We don't need 1 to come along and uglify theorems even further. :yuck:

MathematicalPhysicist
Gold Member
Icebreaker said:
Primes are interesting precisely because 1 isn't a prime. 1 is a divisor of every number, therefore defining 1 as a prime dulls the main purpose of primes.
this is why god=1.

(god-> greatest omnipotent divisor).

a joke, which i hope that number theorists understand. :tongue2:

matt grime
Homework Helper
WeeDie said:
Please tell me, why are primes interesting?
They are interesting because they are devisable by themself and 1 only and I see no reason why you should exclude 1 that definition. The official definition may say something else but until I find a good reason to exclude 1, I will keep it in my list of primes.

That isn't a very good definition of prime and *could* be interpreted to mean 1 isn't prime since 1 is not divisible by 1 *and* itself. Which is why some people prefer to say "exactly two factors" or to be more specific exacty two factors in N, or two positive factors in Z. However, the "real" reason that primes are interesting is that p is a prime if the ideal (p) is a prime ideal, that is p is not a unit and of p|ab then either p|a or p|b. We exclude 1 (and other units) since that special case is not interesting (and we do not allow the whole ring to be considered a prime ideal; prime ideals are proper, though the zero ideal, if prime as it is in a domain, is allowed as a prime ideal, at least in the things I've read recently). Anyway, go around with your different definition of prime then; the only person it will affect will be you.

a prime number is an integer which has exactly 2 distinct factors

Zurtex