• Does anyone know of a proof for this (Conjecture/Theorem): If and only if p is a prime of the form p=1,4 mod 5 and if and only if p has a primitive root g such
Message 1 of 9 , Feb 26, 2001
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Does anyone know of a proof for this (Conjecture/Theorem):

If and only if p is a prime of the form p=1,4 mod 5 and if and only
if p has a primitive root g such that g^2=g+1 mod p then the period
h(p) of F(n) mod p (where F(n) is the nth fibonacci number for
n=1,2,3,...) is given by h(p)=p-1.

Thanks
Louis
• One of our sets of pages on primes is the Prime Curios! which list curiosities for a variety of primes. On the page
Message 2 of 9 , Feb 26, 2001
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One of our sets of pages on primes is the Prime Curios! which list curiosities
for a variety of primes. On the page

http://www.utm.edu/research/primes/curios/first.shtml

we present a false proof that every prime has an associated curio
(included below). This is a play off the old proof "every number is
interesting."

Do any of you know the source of this theorem or a good
reference that discusses the error in the proof? (With
interesting, not Curio, or whatever) Since the idea is not original
with me, I'd like to cite the source.

Chris

The text:

First we need a definition. We will be a little stronger than
Merriam-Webster's definition of curio and make our curios short:

A prime curio about n is a novel, rare or bizarre statement about primes
involving n that can be typed using at most 100 keystrokes.

Theorem: Every positive integer n has an associated prime curio.

Proof: Let S be the set of positive integers for which there is no
associated prime curiosity. If S is empty, then we are done. So suppose,
for proof by contradiction, that S is not empty. By the well-ordering
principle S has a least element, call it n. Then n is the least positive
integer for which there is no associated prime curio. But our last
statement is a prime curio for n, a contradiction showing S does not have a
least element and completing the proof.
• Hiya Chris, I think your archivable ;-) paradox can be credited to Russell, or Burali-Forte. Russell discovered his paradox in May of 1901 while working on
Message 3 of 9 , Feb 27, 2001
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Hiya Chris,

I think your "archivable" ;-) paradox can be credited to Russell, or
Burali-Forte.
"Russell discovered his paradox in May of 1901 while working on his
Principles of Mathematics (1903). Cesare Burali-Forti, an assistant
to Giuseppe Peano, had discovered a similar antinomy in 1897 when he
noticed that since the set of ordinals is well-ordered, it, too, must
have an ordinal. However, this ordinal must be both an element of the
set of ordinals and yet greater than any such element."

It is a simple self-reference with one negation.
Burali-Forti's and your example contains an ordering, "smallest" or
"greatest" which is not important.

Here's a separate Trivial Pursuit question.
Earl Doctor Bertrand Russell FRS OM, the famous Mathematician and
pacifist won the Nobel Prize in 1950. What for?

>
> Russells paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are
> not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.
>
> Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of
> all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself,
> then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox prompted much work in logic, set theory and the philosophy
> and foundations of mathematics during the early part of the twentieth century.
>
> Bibliography
> Other Internet Resources
> Related Entries ...
>

A good site about and including works of Russell is at:-
http://www.humanities.mcmaster.ca/~bertrand/index.html

Cheers,
Paul Landon

"... mathematics is only the art of saying the same thing in different words"
-Bertrand Russell

• Hello! This proof is indeed classical. There is a well-known standard proof of infinity of interesting things (the most non-interesting thing becomes
Message 4 of 9 , Feb 27, 2001
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Hello!

This "proof" is indeed classical. There is a well-known
"standard" proof of infinity of interesting things (the most
non-interesting thing becomes interesting) and of some other
curious things. AFAIK, yet antique philosophers, namely
sophistes (Protagor and others) have invented the most of

Once someone (maybe Russell) simply has decided to apply
such method to prime curios, and so we have this "proof". I
think in this case copyright partially belongs to antique
sophistes. :-)

Hope this will help,

Andrey
--------------------------------------------------
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рп Й вд РПМШЪПЧБФЕМС, РПЮФПЧБС УМХЦВБ! http://tut.by/services.html
• Hi folks ... It is indeed classical, and probably goes back a long long way (long before Russell s formulation). These sort of self-referential statements get
Message 5 of 9 , Feb 27, 2001
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Hi folks

>we present a false proof that every prime has an associated curio
>(included below). This is a play off the old proof "every number is
>interesting."
>Do any of you know the source of this theorem or a good
>reference that discusses the error in the proof?

It is indeed classical, and probably goes back a long long way (long
before Russell's formulation). These sort of self-referential
statements get a lot of treatment in G\"odel, and so the "error in the
proof" is that incompleteness allows us to construct the statement in
the first place.

But that's no fun. I don't have a reference - but I do have a reference
to a reference. Martin Gardner mentions the 'all numbers are
interesting' proof in the first "Mathematical Puzzles and Diversions"
anthology of his Scientific American column. I'm not sure who first
dressed this up for the 'interesting numbers' case, but I'm sure
Gardner includes a reference (Can't say for sure as I don't have the
book to hand).

Chris (N).
• Hi Chrisii, I seem to remember it being in The Penguin Dictionary of Curious and Interesting Numbers , I think as number 6 or 14, but my book is back in
Message 6 of 9 , Feb 28, 2001
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Hi Chrisii,

I seem to remember it being in "The Penguin Dictionary of Curious
and Interesting Numbers", I think as number 6 or 14, but my book
is back in England. This dictionary should reference his sources.

> This is a play off the old proof "every number is interesting."
> >Do any of you know the source of this theorem or a good
> >reference that discusses the error in the proof?

The word "interesting" is not in "Earliest Known Uses of Some of the
Words of Mathematics" at http://members.aol.com/jeff570/mathword.html
and it would be _interesting_ to submit it there once we get to the
bottom of it.

I still think that it is an example of Russell's Paradox similar to
Burali-Forti's example.
mentions [ van Heijenoort, Jean, From Frege to Gödel, Cambridge:
Harvard University Press, 1967 ] many times, so this could be another
reference to a reference.
On that site, it implies that this paradox was not commonly known and
the news of it caused Frege to stop the presses (to add an appendix).

In normal logic you have the symbols 0 and 1, if you have a self-
reference and a negation give this the ordinal 2. Logic Engineers
use the technical terms False, True and Wibblywobbly. With another
self-reference and a negation you can always create another paradox
with ordinal N+1. This system is incomplete, but so what? The counting
numbers are incomplete and there is no largest prime number either,
and this was known in very early times.

Cheers,
Paul Landon
• ... Like Chris, I thought these were rather old, but I spent some time in the library and it turns out that perhaps they are not. The originator seems to be
Message 7 of 9 , Mar 6 8:23 AM
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At 07:47 PM 2/27/01 -0500, Chris Nash wrote:
> >Do any of you know the source of this theorem or a good
> >reference that discusses the error in the proof?
>
>It is indeed classical, and probably goes back a long long way (long
>before Russell's formulation). These sort of self-referential

Like Chris, I thought these were rather old, but I spent some time in
the library and it turns out that perhaps they are not. The "originator"
seems to be G. G. Berry of the Bodleian library who wrote the
paradox (in a messier form) to Russel 21 December 1904. So this
indeed came after the big set of all sets paradox (and the largest
the rest). Russel was the first to publish it in 1906 (Berry never
did). In 1908 Russel published it in this simplified form:

Hence "the least integer not nameable in fewer
than nineteen syllables" must denote a definite
integer; in fact, it denotes 111,777. But "the
least integer not nameable in fewer than
nineteen syllables" is itself a name consisting
of eighteen syllables; hence the least integer
not nameable in fewer than nineteen syllables
can be named in eighteen syllables, which is a

Berry introduced himself to Russell with a card that on one side
said "the statement on the other side of this card is false" and
on the other side said "the statement on the other side of this
card is true"--another (now) well known paradox that he also
originated.

Berry's paradox is not isomorphic to the Curios (or interesting)
paradox, but looks to me like the forerunner.

>statements get a lot of treatment in G\"odel, and so the "error in the
>proof" is that incompleteness allows us to construct the statement in
>the first place.

Russell's set of all sets and the various diagonal arguments
(Richards, G\"odel, ...) in that they can not be directly
expressed within logic. And indeed forms of them are old
(so the Chris' weren't completely off base!) The first is
perhaps Epimenides the Cretan's all Cretan's are liars...
• ... Oh, I bet long before some hunter-gather s grunted along these lines round a fire: Ugg: Caught summat? Glugg: Two no-names! Can t see that the curios thing
Message 8 of 9 , Mar 6 5:35 PM
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Chris Caldwell wrote:

> The first is perhaps Epimenides

Oh, I bet long before some hunter-gather's grunted along
these lines round a fire:

Ugg: Caught summat?
Glugg: Two no-names!

Can't see that the curios thing is much different from
the self-contradictory "Anon" appended to a pome....

David
• ... No doubt. But my context is simpler, I was looking for references; and the first seems to be Epimenides. But I d be glad to reference the hunter
Message 9 of 9 , Mar 7 5:53 AM
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At 01:35 AM 3/7/01 +0000, d.broadhurst@... wrote:
>Chris Caldwell wrote:
> > The first is perhaps Epimenides
>
>Oh, I bet long before some hunter-gather's grunted along
>these lines round a fire:

No doubt. But my context is simpler, I was looking for references;
and the first seems to be Epimenides. But I'd be glad to reference
the hunter gatherers if you know of a cave wall I can cite!

anon.
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