Why is 1 not considered a prime number?

[Loren Booda]

Why is 1 not considered a prime number? It meets the requirement of being only divisible by itself and 1.

 

[AKG]

Because that's not the definition of a prime number. The definition of a prime number is:

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p > 1 that has no positive integer divisors other than 1 and p itself. (Source)

1 certainly does not meet that criteria since 1 is not greater than 1.

 

[honestrosewater]

Couldn't you also say n is prime iff n has exactly two factors? I like that better.

I was just going to ask about the reasons for excluding 1. Does it have something to do with coprimes?

 

[neurocomp2003]

no its just if 1 was a prime number then every other number would be a composite and prime which would defeat the purpose of calling it a prime list.
a prime is suppose to be unique in factorization 1*n...and even though 1 fits the purpose it would destroy the thought and terminology ...but yes you can think of 1 as being prime...its the fundamental number.

 

[matt grime]

That 1 is not prime is purely a convention, and a modern one at that. it makes more sense for it not to be a prime. it isn't a composite either, it is a unit.

this kind of question, to my mind, fits in with the ones i get asked a lot like: but why do groups satisfy those 4 axioms. it's almost as if people believe that the axioms we choose are somehow god given, carved in some stone and we must make sense of these mysterious rules that came from nowhere when in fact they are man made.

 

[MathematicalPhysicist]

this kind of question, to my mind, fits in with the ones i get asked a lot like: but why do groups satisfy those 4 axioms. it's almost as if people believe that the axioms we choose are somehow god given, carved in some stone and we must make sense of these mysterious rules that came from nowhere when in fact they are man made.

i don't see any problem with god given axioms espcecially when "god" itself is man made definition.

 

[matt grime]

some axioms are more believable than others (see the axiom of choice for instance for one that isn't) but there no absolute truths. we study, say, groups, not becuase someone one day from absolutely nowehre said ooh, these four axioms i wonder... but because the study of certain objects over time were unified as it was observed that they had common properties.

 

[Icebreaker]

1 being prime would invalidate the fundamental theorem of arithmetic.

 

[matt grime]

invalidate? not the word i'd've chosen but then that may just be me being picky. the statement of the theorem is dependent upon us accepitng the definitions properly.

 

[shmoe]

no its just if 1 was a prime number then every other number would be a composite and prime which would defeat the purpose of calling it a prime list.

This isn't true. Remove the "p>1" clause from the definition in AKG's post and the only change is 1 is now considered a prime (assuming you aren't going to quibble over the "1 and p itself" bit). It has no effect on any other number being prime or not.



It's just a convention that we use this definition. It has no effect on the mathematics behind any theorems, only in how we state them i.e. the fundamental theorem of arithmetic would not suddenly be wrong just irritating to state. It's turned out to be convenient to separate primes from the units, so we build the definitions to take this into account.

 

[Loren Booda]

From the above posts, it would seem that Occam's razor would mandate as extraneous the separate condition "p>1" for primes. honestrosewater's definition

n is prime iff n has exactly two factors

includes this condition, thus effecting parsimony.

 

[WeeDie]

I think it's kinda stupid not to see 1 as a prime.

 

[Daminc]

How long has 1 not been considered a prime?

 

[arildno]

I think it's kinda stupid not to see 1 as a prime.

Why?

How long has 1 not been considered a prime?

From the beginning of the 19th century, I would believe, since that was approximately the time when mathematicians realized the need to take more care in what definitions they chose to use.

 

[Daminc]

I see, it's such a long time ago that I learned about primes that I couldn't remember if 1's were included then. Thanks for the clarification.

 

[WeeDie]

arildo: because the characteristic that makes primes interesting is the fact that they are only devisable by themself and 1. The number 1 serves this condition so I see no need to exclude 1 from the definition of primes. In fact, one might get off on a bad start if one were to exclude 1 and graph primes, in order to find connections between primes and other number series.

 

[arildno]

No, the characterestic of a prime is that the whole number is greater than 1 and is divisible only with itself and 1. What you may find "interesting" is of little importance.

 

[matt grime]

arildo: because the characteristic that makes primes interesting is the fact that they are only devisable by themself and 1. The number 1 serves this condition so I see no need to exclude 1 from the definition of primes. In fact, one might get off on a bad start if one were to exclude 1 and graph primes, in order to find connections between primes and other number series.

that isn't the proper definition of prime, it is your vwersin of the definition. and in any case it is better stated as "has exactrly two positive factors" since this precludes 1 (and even characterizes primes in the integers as well as the naturals).

 

[arildno]

that isn't the proper definition of prime, it is your vwersin of the definition. and in any case it is better stated as "has exactrly two positive factors" since this precludes 1 (and even characterizes primes in the integers as well as the naturals).

I assume that was directed at me.
Thanks for the correction.

 

[HallsofIvy]

I think it's kinda stupid not to see 1 as a prime.

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...

 

[EL]

it is better stated as "has exactrly two positive factors" since this precludes 1 (and even characterizes primes in the integers as well as the naturals).

How do you get that the primes must be naturals out of this definition?

To me it seems like e.g. -7 is a prime from this (factors -7,-1,1,7...exactly two positive factors...)

 

[matt grime]

erm yes i did just implicitly state the -7 is a prime in Z, and that is perfectly correct and is what the real definition of prime in an arbitrary ring tells us about primes in Z.

 

[EL]

erm yes i did just implicitly state the -7 is a prime in Z, and that is perfectly correct and is what the real definition of prime in an arbitrary ring tells us about primes in Z.

Ah, I misunderstood your sentence. Thought you ment that your definition restricted the primes to be naturals, but now I see...(I can blame my bad English...).
I should have known better than trying to point out a mistake by the Math Guru!

 

[NewScientist]

A prime number is an integer that has exactly two distinct factors.

 

[EL]

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...

How about the positive integer "1" then?

 

[NewScientist]

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.

 

[master_coda]

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.

 

[NewScientist]

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?!

"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!

 

[AKG]

How about the positive integer "1" then?

Let p_i be the i^th 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]

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]

Let p_i be the i^th 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...

 

[WeeDie]

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.

 

[shmoe]

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.

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.

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]

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.