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Topic: 1 as a prime number
Replies: 11   Last Post: Dec 9, 1997 11:13 AM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: 1 as a prime number
Posted: Dec 5, 1997 5:28 PM
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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

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