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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 9, 1997 11:13 AM
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On Tue, 9 Dec 1997, mark snyder wrote:
> >>
> > I answered this at some length a few days ago. The tradition
> >before this century (dating back to Euclid's Elements) was indeed
> >to count 1 as a prime, and even Lehmer's 1914 list of prime numbers to
> >10 million does so. But the many inconveniences it causes have
> >led people this century to put it into a new category, and call
> >it a "unit" rather than a prime.
> >
> > John Conway

> And I appreciate the time you took to answer my question. Perhaps I was
> not clear on what the question was. I know why it is undesirable to
> consider 1 as a prime number, and it is interesting that it was considered
> prime by Lehmer as late as 1914. But my question was: when did it became
> gradually accepted that we should not consider 1 as a prime number? A
> colleague of mine had said that he read somewhere (but didn't remember
> where) that 1800 was some kind of watershed in this regard, and what I was
> hoping for was some reference where I might read more about the history of
> this, hence the post to the math history list. Evidently my colleague was
> off by 100 years, but I would still be interested in any references.
> And my comment about the "big meeting" was my (evidently feeble) attempt at
> jocularity...
> mark snyder

No apology's needed - I thought it might be jocular, but since you
might really have thought that perhaps there was a decision to change
at some meeting, pointed out that there wasn't.

I'm not sure that you saw my first reply, which was considerably
more detailed than the one above. The changeover has been very
gradual, and I'll bet there are publications from the last few years
in which 1 is still counted as a prime - in other words, it's not
yet complete. In the 1950s and 1960s, books that chose the new definition
would always be careful to point out that they were doing so, and that
most authors included 1 with the primes.

The real thing that gets such a change accepted is when it gets
into high-school textbooks. I think that perhaps we must thank "the
new math" movement, which for all its faults did get some of the
terminology and conventions into the high schools that had hitherto
only been used in the Universities. School textbooks don't like to
muddy the waters by explaining such things as variations in usage,
so would tend to give just one definition. My guess is that you'll
find that schoolbooks of the 1950s defined primes so as to include 1,
while those of the 1970s explicitly excluded 1.

John Conway

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