Re: Is this proof of infinitely many primes flawed?

On 30 Jan, 20:31, quasi <qu...@xxxxxxxx> wrote:
On 30 Jan 2009 14:56:26 -0500, Bill Dubuque <w...@xxxxxxxxxxxxxxxxxxxx>
wrote:





quasi <qu...@xxxxxxxx> wrote:
David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:

I didn't make any claims about what it was or was
not "likely" that the OP was asking, just about what
he did in fact ask:

"how can we say p_k  divides evenly (p_1,p_2,...,p_n)?"

There is after all a context.

Yes. The context _seems_ to be that he was confused,
thinking that the answer to his (trivial) question had
something to do with that other assumption, which
it doesn't.

Actually, I think the confusion might in fact stem from the use of
commas instead of a standard product symbol.

Of course, it makes no sense at all to think that p_k would divide
evenly into a _list_ of distinct primes, so _we_ (on sci,math) just
assumed (knowing how the correct proof goes) that the comma was
intended as a product symbol. But the OP (who may simply have copied
it incorrectly off the board in class), may have interpreted the
expression (p_1,p_2,...,p_n) as a list, hence the puzzlement.

That notation might in fact denote the LCM of the p_i
if the teacher attempted to present the proof in a form
somewhat faithful to Euclid's original proof.

I doubt it.

Most likely the teacher wrote the product as

   p_1*p_2 ... *p_n

but used a  "middle dot" for multiplication, as well as the "lower
dots" shown above for continuation. Unfortunately, the student, not
understanding the proof at the time, and copying it down quickly,
probably wrote the "middle dots" as commas. Moreover, earlier in the
proof, there probably _was_ a list of primes separated by commas --
something like

   "Let p_1, p_2, ..., p_n be a list of all the primes"

so the student may have assumed (haste plus lack of understanding)
that the new expression was the same thing. Then, looking at it later,
the student couldn't make any sense of it.

You might be forgetting the crucial second posting
from the OP.

But in supposing p_k | M doesn't that
contradict the "clearly p_k | p_1,p_2,...,p_n"
part?

Why do I say that? Because M is p_1,p_2,...,p_n + 1
note the + 1.

It's clear that he means M is the product of
all the primes assumed ro exist plus one.
He has found a contradiction "before the author
of the original proof does".
The only real difficulty is "supposing".
One of the n primes available must divide M
as M is greater than the last prime, i.e. M can't be
a unit. It is not a supposition.
.

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