Prime numbers

[medico]

Having just read A curious Incident of the Dog in the Night, why does 1 not appear as a prime number? Have heard a couple of arguments, but would be pleased to have ur views!

[Pickle]

Excellent book! Anyway, a number is prime if it has only two factors, 1 and itself. 1 has only 1 factor so it can't be a prime number.

[Jabba]

Prime numbers are by definition greater than 1. The reason for this lies in a technical result known as The Fundamental Theorem Of Arithmetic. Unless you are studying maths, I would be content with knowing that 1 is not prime.

[bernardo]

Another way of putting the same reason as mentioned above is that each compound number (i.e. each non-prime number) can be expressed in only one way as the product of prime numbers. If you count 1 as a prime number, you could do it in more than one way,
e.g. 12=2 x 2 x 3, but also
12 = 1 x 2 x 2 x 3 or
12 = 1 x 1 x 1 x 1 x 1 x 2 x 2 x 3 etc.

[jenstar]

Interestingly (well, to me, anyway), I've been reading a few old maths books from the 1950s and 60s recently, and they tend to say that 1 IS a prime number. Of course, it kinda screws up Erastosthenes' sieve if you treat 1 as prime...

[humming bird]

From Web site of Chris Caldwell at UTM. Note a lot more info is there...type in "is one a prime number" into google. This is a short help. There was a time that many folks defined one to be a prime, but it is the importance of units and primes in modern mathematics that causes us to be much more careful with the number one (and with primes). When we only consider the positive integers, the role of one as a unit is blurred with its role as an identity; however, as we look at other number rings (a technical term for systems in which we can add, subtract and multiply), we see that the class of units is of fundamental importance and they must be found before we can even define the notion of a prime. For example, here is how Borevich and Shafarevich define prime number in their classic text "Number Theory:" An element p of the ring D, nonzero and not a unit, is called prime if it can not be decomposed into factors p=ab, neither of which is a unit in D. Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab, must divide a or b (these classes are the same for the ordinary integers--but not always in more general systems). Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes. Cheers

[Brugel]

Euclid rejected 1 as a prime number because all the (other) primes had the property that the sum of their divisors is one more than the prime itself. To avoid having an exception to this rule, he therefore rejected 1 as a prime.

[Brugel]

Sorry, I meant to say Euler- not Euclid. Euclid, like most of the Ancient Greeks, didn't, I think, regard 1 as a number at all. So the problem of expressing a compound number as the product of primes in more than one way didn't arise since they wouldn't have included 1 in the product anyway.

[Canib]

Brugel, I think that you are mixing in the proof that the number of primes is infinite. If we know all primes to x, then, if you multiply all of them together and add one, that must be prime. Anyway, that is what your post sounded a bit like.

[medico]

Thanks, Guys...thought that one might stir a debate. Some very cogent answers there. I am edified.