- #1

- 3,077

- 3

## Main Question or Discussion Point

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

- Thread starter Loren Booda
- Start date

- #1

- 3,077

- 3

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

- #2

AKG

Science Advisor

Homework Helper

- 2,565

- 4

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

- #3

honestrosewater

Gold Member

- 2,105

- 5

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

- #4

- 1,356

- 2

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

matt grime

Science Advisor

Homework Helper

- 9,395

- 3

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

MathematicalPhysicist

Gold Member

- 4,285

- 200

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

- #7

matt grime

Science Advisor

Homework Helper

- 9,395

- 3

- #8

Icebreaker

1 being prime would invalidate the fundamental theorem of arithmetic.

- #9

matt grime

Science Advisor

Homework Helper

- 9,395

- 3

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:

- #10

shmoe

Science Advisor

Homework Helper

- 1,992

- 1

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.neurocomp2003 said: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.

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

- 3,077

- 3

includes this condition, thus effecting parsimony.n is prime iff n has exactly two factors

- #12

- 12

- 0

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

- #13

- 36

- 0

How long has 1 not been considered a prime?

- #14

arildno

Science Advisor

Homework Helper

Gold Member

Dearly Missed

- 9,970

- 132

Why?WeeDie said:I think it's kinda stupid not to see 1 as 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 said:How long has 1 not been considered a prime?

- #15

- 36

- 0

- #16

- 12

- 0

- #17

arildno

Science Advisor

Homework Helper

Gold Member

Dearly Missed

- 9,970

- 132

- #18

matt grime

Science Advisor

Homework Helper

- 9,395

- 3

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).WeeDie said:

- #19

arildno

Science Advisor

Homework Helper

Gold Member

Dearly Missed

- 9,970

- 132

I assume that was directed at me.matt grime said: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).

Thanks for the correction.

- #20

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 955

Do you consider the Fundamental Theorem of Arithmetic:WeeDie said:I think it's kinda stupid not to see 1 as a prime.

"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= 1

- #21

EL

Science Advisor

- 547

- 0

How do you get that the primes must be naturals out of this definition?matt grime said: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).

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

- #22

matt grime

Science Advisor

Homework Helper

- 9,395

- 3

- #23

EL

Science Advisor

- 547

- 0

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...).matt grime said:

I should have known better than trying to point out a mistake by the Math Guru!

Last edited:

- #24

- 167

- 0

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

- #25

EL

Science Advisor

- 547

- 0

How about the positive integer "1" then?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= 1^{2}*2*3 or...

- Last Post

- Replies
- 25

- Views
- 11K

- Last Post

- Replies
- 3

- Views
- 3K

- Replies
- 3

- Views
- 4K

- Replies
- 47

- Views
- 12K

- Last Post

- Replies
- 10

- Views
- 3K

- Replies
- 5

- Views
- 8K

- Last Post

- Replies
- 1

- Views
- 2K

- Replies
- 3

- Views
- 4K

- Last Post

- Replies
- 7

- Views
- 2K

- Last Post

- Replies
- 1

- Views
- 2K