Why is 1 not considered a prime number?

1. Jul 2, 2005

Loren Booda

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

2. Jul 2, 2005

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.

3. Jul 2, 2005

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?

4. Jul 2, 2005

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.

5. Jul 2, 2005

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.

6. Jul 2, 2005

MathematicalPhysicist

i dont see any problem with god given axioms espcecially when "god" itself is man made definition.

7. Jul 2, 2005

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.

8. Jul 2, 2005

Icebreaker

1 being prime would invalidate the fundamental theorem of arithmetic.

9. Jul 2, 2005

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.

Last edited: Jul 2, 2005
10. Jul 4, 2005

shmoe

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 seperate primes from the units, so we build the definitions to take this into account.

11. Jul 4, 2005

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
includes this condition, thus effecting parsimony.

12. Aug 23, 2005

WeeDie

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

13. Aug 23, 2005

Daminc

How long has 1 not been considered a prime?

14. Aug 23, 2005

arildno

Why?
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.

15. Aug 23, 2005

Daminc

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

16. Aug 23, 2005

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.

17. Aug 23, 2005

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.

18. Aug 23, 2005

matt grime

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).

19. Aug 23, 2005

arildno

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

20. Aug 23, 2005

HallsofIvy

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...