
Re: 1 as a prime number
Posted:
Dec 5, 1997 5:28 PM


On Fri, 5 Dec 1997, mark snyder wrote:
> I have heard that until about 1800, 1 was considered to be a prime number. > Is this correct? At what point was a decision made to define 1 as not > prime? Was there some sort of big meeting at which this was decided, or > some work which is seminal in this regard? > > mark snyder > The change gradually took place over this century, because it simplifies the statements of almost all theorems. If you count 1 as a prime, for example, numbers don't have unique factorizations into primes, because for example 6 = 1 times 2 times 3 as well as 2 times 3. It's a bit of a nuisance that Lehmer's 1914 "List of all prime numbers below 10 million" counts 1 as a prime.
There was no big meeting, just a gradual consensus of opinion. I think the development of number theory for other rings played a big part, because there one finds other "units" besides 1 (for instance +1 and +i in the Gaussian integers), and these units clearly behave in many ways that make them different from the primes.
Other examples of the kind of thing that goes wrong if you count 1 as a prime are arithmetical theorems like
"If p,q,r,... are distinct primes, then the number of divisors of p^a.q^b.r^c.... is (a+1)(b+1)(c+1)... ."
Mathematicians this century are generally much more careful about exceptional behavior of numbers like 0 and 1 than were their predecessors: we nowadays take care to adjust our statements so that our theorems are actually true. It's easy to find lots of statements in 19th century books that are actually false with the definitions their authors used  one might well find the above one, for instance, in a work whose definitions allowed 1 to be a prime. Nowadays, we no longer regard that as satisfactory.
John Conway

