PDA

View Full Version : Is one a prime number?

copperwindow
09-03-2009, 11:43 PM
Why or why not?

tr0psn4j
09-03-2009, 11:55 PM
According to wikipedia, no. (http://en.wikipedia.org/wiki/Prime_number) It needs two distinct divisors. So 2 is the first prime number.

Colibri
09-03-2009, 11:57 PM
No, by definition. (http://en.wikipedia.org/wiki/Prime_number)

A natural number is called a prime, a prime number or just prime if it has exactly two distinct divisors. Otherwise it is called composite. Therefore, 1 is not prime, since it has only one divisor, namely 1.

The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any n = n x 1. In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable."

Bolding mine.

ultrafilter
09-03-2009, 11:59 PM
No. Integer primes are required to have exactly two positive divisors. One only has itself.

The alternate, more general definition allows a number p to be a prime whenever two conditions are met:

There's no product ab such p divides ab with zero remainder, but p does not divide a and p does not divide b.
There's no number q such that pq = 1.

One satisfies the first requirement, but clearly not the second.

Note that one is also not composite. It falls into a separate class known as units, which are those integers that have reciprocals. In the positive integers, one is the only unit, but in the unrestricted integers negative one is also a unit. Just for completeness, let me mention that zero falls in a fourth category, the so-called zero divisors.

Tapioca Dextrin
09-04-2009, 12:39 AM

On a tangential topic, 1 is usually excluded from the list of prime numbers. One seems to fit the definition of prime number ("having no divisors except itself and 1"); but to include 1 as a prime number would eliminate the unique factorization theorem, that every number can uniquely be expressed as a product of prime numbers. Hence, 1 is not considered to be a prime number.

Any number not prime is called composite, and 1 is also excluded from the composite numbers. So, in effect, you have 1 is neither prime nor not-prime. Maybe that's what stirred your memory when you were thinking of zero as neither even nor not-even.

Not-even, of course, is a mouse.

jackelope
09-04-2009, 12:48 AM
Note that one is also not composite. It falls into a separate class known as units, which are those integers that have reciprocals.I'm not a math person (and you obviously are), but did you mean to say that units are those numbers that are their own reciprocals? Or those numbers that have reciprocals that are also integers? Doesn't every number have a reciprocal?

Indistinguishable
09-04-2009, 12:50 AM
The alternate, more general definition allows a number p to be a prime whenever two conditions are met:

There's no product ab such p divides ab with zero remainder, but p does not divide a and p does not divide b.
There's no number q such that pq = 1.

One satisfies the first requirement, but clearly not the second.
In case it's not clear why one would have both these conditions, note that taken together, they are equivalent to the single clean condition:

Whenever p divides a product of finitely many numbers, then p divides one of those numbers.

The first condition above is the case for binary products, the second condition is the case for nullary products, and, as usual, these two cases suffice to cover everything else as well.

On edit, to jackelope: Units (in some ring, or whatever context) are those numbers that have reciprocals (in that ring, or whatever context). So within the context of the integers, the units are those integers which have integer reciprocals. Within the context of something else, the units are those whatever elses that have reciprocals that are whatever elses (as the context is one in which we are only talking about whatever elses). In some contexts, everything (non-zero) has a reciprocal (e.g., real numbers), but in many other contexts (e.g., linear transformations upon two-dimensional space, with addition given pointwise and multiplication given by composition, for one natural, if mathy, example) many things don't.

jackelope
09-05-2009, 04:51 PM
Thanks, Indistinguishable. That makes sense.

justrob
09-05-2009, 05:17 PM
Interesing. I always thought 1 was a prime number because I thought the definition of a prime number is any number that can only be divided by 1 or itself without any remainder. 1 passes that test if you don't mind that 1 is being divided by 1 in both cases.

Of course I'm no mathematician or really all that good at math at all so there you go.

09-06-2009, 01:51 AM
Nit pick: That was a staff report, and not from Cecil.

I literally just got up and am not so swift this morning, so I am not getting the joke: Not-even, of course, is a mouse. Can anyone fill me in?

TWDuke
09-06-2009, 02:02 AM
I literally just got up and am not so swift this morning, so I am not getting the joke: Not-even, of course, is a mouse. Can anyone fill me in?"Not-Even, a mouse" equals "not even a mouse (http://www.bartleby.com/248/27.html)"?

09-06-2009, 03:06 AM
"Not-Even, a mouse" equals "not even a mouse (http://www.bartleby.com/248/27.html)"?
:D Thank you.

Hari Seldon
09-06-2009, 06:08 PM
Just for the record, 0 is definitely an even number. An even number is, by definition, twice some number and 0 = 2*0 eminently qualifies.

ultrafilter
09-06-2009, 06:33 PM
One other comment on the article: Dex is taking the natural numbers to be {1, 2, 3, ...}, but a lot of people will regard the natural numbers as {0, 1, 2, 3, ...}.

09-08-2009, 12:13 AM
The reason that 1 is not prime has nothing to do with the number of divisors!

1 is a unit in that it can be multiplied be another number (in this case, itself 1) to equal 1. By definition, a unit cannot be prime.

Pleonast
09-08-2009, 10:46 AM
One is not a prime number by definition. What needs to be asked is "why does the definition of prime excludes one?". The answer to that is that we wish every natural number to have a unique prime factorization. That's not possible if we can add any number of ones onto the factorization.

For example, without a prime one, 60 = 22 x 3 x 5. That's unique. If one is prime, we also have 60 = 1 x 22 x 3 x 5 = 12 x 22 x 3 x 5, etc. There's no unique factorization.

Thudlow Boink
09-08-2009, 11:54 AM
I suspect a lot of other mathematical theorems would have to be rewritten, with "Let p be a prime" changed to "Let p be a prime greater than 1."

Lance Turbo
09-08-2009, 08:38 PM
The reason that 1 is not prime has nothing to do with the number of divisors!

1 is a unit in that it can be multiplied be another number (in this case, itself 1) to equal 1. By definition, a unit cannot be prime.

Are you joking? I can't imagine why someone would say this if he or she wasn't joking.

Some ring theoretical notions of prime elements or prime ideals might not mention the number of divisors but that's not being discussed here. The topic of this thread is prime numbers and the the number theoretical definition of such certainly does mention the number of divisors.

A prime number is a positive integer p > 1 that has no positive integer divisors other than 1 and p itself.

copperwindow
09-09-2009, 01:56 AM
Are you joking? I can't imagine why someone would say this if he or she wasn't joking.

Some ring theoretical notions of prime elements or prime ideals might not mention the number of divisors but that's not being discussed here. The topic of this thread is prime numbers and the the number theoretical definition of such certainly does mention the number of divisors.

Wendell Wagner
09-09-2009, 02:11 AM
This should work:

The problem is that if you have a parenthesis at the end of a URL, even if it's not supposed to be part of the URL, it screws up things. Indeed, if a link doesn't work, I would suggest that posters try messing around with things like removing and moving parentheses rather than immediately complaining that it doesn't work. Usually the problem is something small like this.

Chronos
09-09-2009, 12:08 PM
The problem is that if you have a parenthesis at the end of a URL, even if it's not supposed to be part of the URL, it screws up things.Unless, of course, it is supposed to be part of the URL, in which case it often seems to screw up the other way.

Contrapuntal
09-09-2009, 12:13 PM
This should work: