## Caldwell conjectured #Wieferich primes = finite?

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• What is the reasoning behind this conjecture? (Wikipedia says Caldwell conjectured this.)
Message 1 of 8 , Mar 26, 2013
What is the reasoning behind this conjecture?
(Wikipedia says Caldwell conjectured this.)
• ... I incline to the Crandall-Pomerance heuristic that there is an infinitude, with O(log(log(N))) less than N. (CP Section 1.3.3.) David
Message 2 of 8 , Mar 26, 2013

> What is the reasoning behind this conjecture?

I incline to the Crandall-Pomerance heuristic
that there is an infinitude, with O(log(log(N)))
less than N. (CP Section 1.3.3.)

David
• ... Me too. I have also been planning to ask Chris. In fact I have a whole file with comments to the Prime Pages. I should probably get around to actually
Message 3 of 8 , Mar 26, 2013
David wrote:
> I incline to the Crandall-Pomerance heuristic
> that there is an infinitude, with O(log(log(N)))
> less than N. (CP Section 1.3.3.)

Me too. I have also been planning to ask Chris. In fact I have a whole file
with comments to the Prime Pages. I should probably get around to actually
mailing it to him one day before it all becomes obsolete. Below is what it has
said for a long time.

http://primes.utm.edu/glossary/xpage/WieferichPrime.html says:
"Are there infinitely many Wieferich primes? Probably not"

I wonder what "Probably not" is based on. The Crandall, Dilcher and Pomerance
reference says on page 14 of
http://www.math.dartmouth.edu/~carlp/PDF/paper111.pdf:
"Thus, heuristically we might argue that the number of Wieferich primes in an
interval [x; y] is expected to be sum(1/p) over x<=p<=y ~= ln(ln y/ln x)."

This is the same as the estimate on
http://primes.utm.edu/glossary/xpage/WilsonPrime.html
(The reason for the estimates is the same: A guess that each p has chance 1/p)

--
Jens Kruse Andersen
• First, Caldwell (if that means me) made no such conjecture. Folks are too lose with that word. Why I put the opinion probably not on that page some years
Message 4 of 8 , Mar 26, 2013
First, Caldwell (if that means me) made no such "conjecture." Folks are too lose with that word. Why I put the opinion "probably not" on that page some years ago, I have no clue now myself. The reference I linked to that page suggests the opposite (also carefully not stated as a conjecture) which seems to suggest I intended "probably."

-----Original Message-----
Sent: Tuesday, March 26, 2013 1:40 PM
Subject: [PrimeNumbers] Caldwell conjectured #Wieferich primes = finite?

What is the reasoning behind this conjecture?
(Wikipedia says Caldwell conjectured this.)

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• ... And if we re being loose with the word, and accepting that there s an enormous, and unmeasurable, level of doubt in such statements, then it s not uncommon
Message 5 of 8 , Mar 27, 2013
--- On Wed, 3/27/13, Chris Caldwell <caldwell@...> wrote:
> First, Caldwell (if that means me)
> made no such "conjecture."  Folks are too lose with
> that word.  Why I put the opinion "probably not" on
> that page some years ago, I have no clue now myself.
> The reference I linked to that page suggests the opposite
> (also carefully not stated as a conjecture) which seems to
> suggest I intended "probably."

And if we're being loose with the word, and accepting that there's an enormous, and unmeasurable, level of doubt in such statements, then it's not uncommon to come up with paradoxical conclusions.

Let's look also at the another set with the same expected density, but even fewer are known - the \$620 pseudoprimes. A conjecture about the density of those grossly overestimate the number that we've actually found - none. What are the chances of there being so few? The heuristics are simple, but the heuristics are naive. The data is simple, but the data is unassailably true.

It's not a big leap to say "one of the assumptions behind the heuristic must be wrong". Probably.

Which of course if different from a statement that the tally of such objects is finite - it only takes a small factor to make something infinite become pointlessly sparse.

Phil
--
() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]
• ... Indeed. It was an obiter dictum , now changed in http://primes.utm.edu/glossary/xpage/WieferichPrime.html to the converse remark. Neither then nor now
Message 6 of 8 , Mar 27, 2013
Chris Caldwell <caldwell@...> wrote:

> First, Caldwell (if that means me) made no such "conjecture."
> Folks are too lose with that word.

Indeed. It was an "obiter dictum", now changed in
http://primes.utm.edu/glossary/xpage/WieferichPrime.html
to the converse remark. Neither then nor now should Warren,
or others, seek to elevate it to a "conjecture" by Chris.

I have found Warren's contributions fruitful enough to set
aside his unfortunate tone in communicating them. Yet I am
still puzzled why he seems to need to be so combative in
what is, quite frankly, a place where he should not expect
the type of serious review that he might gain by taking the
trouble to write up his ideas in an appropriate form for
scrutiny by a more rigorous mathematical community.

In any case, thanks Warren, for the maths.

David (not a mathematician)
• ... --I just saw it in wikipedia, not elsewhere. I did not elevate it, I simply reported what wikipedia said, and asked about it, in a concise way. ...
Message 7 of 8 , Mar 27, 2013
>
>
>
> Chris Caldwell <caldwell@> wrote:
>
> > First, Caldwell (if that means me) made no such "conjecture."
> > Folks are too lose with that word.
>
> Indeed. It was an "obiter dictum", now changed in
> http://primes.utm.edu/glossary/xpage/WieferichPrime.html
> to the converse remark. Neither then nor now should Warren,
> or others, seek to elevate it to a "conjecture" by Chris.

--I just saw it in wikipedia, not elsewhere. I did not "elevate" it,
a concise way."

> I have found Warren's contributions fruitful enough to set
> aside his unfortunate tone in communicating them. Yet I am
> still puzzled why he seems to need to be so combative in

--i am not being "combative." I was being "inquisitive."
YOU, however, are being combative. Right now.

> what is, quite frankly, a place where he should not expect
> the type of serious review that he might gain by taking the
> trouble to write up his ideas in an appropriate form for
> scrutiny by a more rigorous mathematical community.
>
> In any case, thanks Warren, for the maths.
>
> David (not a mathematician)

--look David, you seem to make a habit of complaining about my "tone".
But your usual complaint-method omits actually explaining what the problem is.
In the rare cases where you do explain, such as here... I find it unjustified.

Here, for reference, was my entire damn post, in full, verbatim:
"What is the reasoning behind this conjecture?
(Wikipedia says Caldwell conjectured this.)"

end. full stop. I am satisfied with the answer I got to this enquiry (though I was expecting something deeper :)

Quite frankly, I find it puzzling why David seems to need constantly to be so
worried about others' tone, and/or why -- given that he does feel that need --
he feels the urge to be so mysterious about it when he does so.
Indeed, the very word "tone" is a very mysterious one. Anyhow, let me conjecture
that, similarly to the way some appreciate Beethoven's symphonies, while others who
are "tone deaf" (also called "amusic") hear nothing but somewhat annoying noise --
I do not perceive a vast symphony of something, that apparently is so obvious to David Broadhurst, that he cannot stop complaining about it, but at the same time it is so obvious to him that he sees no reason to actually explain instances.

Perhaps Broadhurst should take this conjecture into account before/during
his issuings of future complaints. I would assume the amusic are annoyed at being subjected to Beethoven, but even more annoyed about being harangued about it by Beethoven-lovers.
• ... Twice (on this list) is hardly a habit. Phil has done so once: http://tech.groups.yahoo.com/group/primenumbers/message/24018 ... Thanks again, Warren, for
Message 8 of 8 , Mar 28, 2013
"WarrenS" <warren.wds@...> wrote:

> --look David, you seem to make a habit of complaining about
> my "tone".

Twice (on this list) is hardly a habit.

Phil has done so once: