Finite sets of primes

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• ... Aside from wholly trivial cases which Chris Caldwell has partly covered (even primes, n-digit primes, primes a
Message 1 of 9 , Oct 1, 2003
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At 01:33 PM 10/1/2003 +0000, mad37wriggle wrote:

>Are there any types/forms of primes of which there are definitely
>only a finite number? I'm sure there must be, but can't think of
>any off the top of my head. I'd be interested to see the proofs of
>finitude...

Aside from wholly trivial cases which Chris Caldwell has partly covered
(even primes, n-digit primes, primes a < x < b where x is prime, yadda yadda
involve dragging in boundary conditions from disciplines other than number
theory. For example: group theory. As we all know, Galois proved that
algebraic equations > degree 5 have roots which cannot be found by
Cardano-type cubic or quadratic closed formulas. So all primes < 5 fall into
a special class for that reason.
Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
using compass and straightedge, so all primes < 17 also fall into another
special class. The same idea could doubtless be carried out by keelhauling
a wide variety of disciplines into forcible contact with the primes. Viz.,
algebraic topology: 4-manifolds are known to be peculiarly difficult to find
minimal surfaces in, if memory serves; physics: a real treasure trove of
arbitrarily large but finite sets, which would include subsets of primes --
close packing of various regular solids offer only limited solutions
including subsets of primes; solid geometry: the Gauss formula for vertices,
etc.
By dragging in other disciplines, an unbounded number of sets of finite
primes could doubtless be classified. The trick involves finding boundary
conditions unrelated to pure number theory, but calling on different realms
of mathematics such as plane gemoetry, statistical mechanics, topology, ad
infinitum.
Even within number theory, we can easily define an ubounded number of sets
of finite groups of primes. For example: the number of primes < a hailstone
number run for a given seed. Or the number of primes less than each perfect
number as we successively march upward. Or the number of primes less than
the multiple-valued logs of any given imaginary number. Und so weiter.
• ... Up to 1.25*10^15 there are only two Wieferich primes: 1093 and 3511. Prime p is a Wieferich prime if 2^(p-1) = 1 (mod p^2). But there isn t a prove that
Message 2 of 9 , Oct 1, 2003
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At 01:33 PM 10/1/2003 +0000, mad37wriggle wrote:

> Are there any types/forms of primes of which there are definitely
> only a finite number? I'm sure there must be, but can't think of
> any off the top of my head. I'd be interested to see the proofs of
> finitude...

Up to 1.25*10^15 there are only two Wieferich primes: 1093 and 3511. Prime p
is a Wieferich prime if 2^(p-1) = 1 (mod p^2).

But there isn't a prove that there are finitely many Wieferich primes.
Best regards,

Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosa@...
• ... Gauss proved that it could be constructed.
Message 3 of 9 , Oct 1, 2003
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On Wed, 1 Oct 2003, xed wrote:
> Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
> using compass and straightedge

Gauss proved that it could be constructed.
• ... Not only is there not a proof, heuristic arguments suggest that there should be an infinite number of them. Paul
Message 4 of 9 , Oct 1, 2003
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> Up to 1.25*10^15 there are only two Wieferich primes: 1093
> and 3511. Prime p
> is a Wieferich prime if 2^(p-1) = 1 (mod p^2).
>
> But there isn't a prove that there are finitely many Wieferich primes.

Not only is there not a proof, heuristic arguments suggest that there should be an infinite number of them.

Paul
• ... An n-gon is constructible (with usual rules) if and only if it is a product of a power of two and distinct Fermat primes. So 17 gets us back to the
Message 5 of 9 , Oct 1, 2003
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At 11:26 AM 10/1/2003 -0400, Carl Devore wrote:

>On Wed, 1 Oct 2003, xed wrote:
>> Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
>> using compass and straightedge
>
>Gauss proved that it could be constructed.

At 11:26 AM 10/1/2003 -0400, you wrote:
>On Wed, 1 Oct 2003, xed wrote:
>> Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
>> using compass and straightedge
>
>Gauss proved that it could be constructed.

An n-gon is constructible (with usual rules) if and only if it is a product of a power of two and
distinct Fermat primes. So 17 gets us back to the Fermats which might be finite... but...

Chris

(And indeed I left out 2 with x^n-1. The idea is to take any polynomial that factors,
it will be prime only if at all but one of the factors is a unit (2 make x-1 a unit). But
that does give an infinite number of such trivial examples.)
• ... May be within number theory, but are still somewhat artificial examples. It s all just choosing a particular upper bound, and taking all primes less than
Message 6 of 9 , Oct 3, 2003
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> Even within number theory, we can easily define an ubounded number of sets
> of finite groups of primes. For example: the number of primes < a hailstone
> number run for a given seed. Or the number of primes less than each perfect
> number as we successively march upward. Or the number of primes less than
> the multiple-valued logs of any given imaginary number. Und so weiter.

May be within number theory, but are still somewhat artificial examples.
It's all just choosing a particular upper bound, and taking all primes
less than it. That's nothing particularly special, no matter where the
upper bound comes from.

But if looking at finite sets, it does no harm to consider all integers,
not just primes. And the class number problem provides a good example
for the original question posed...

Let D<0 be the discriminant of an imaginary quadratic field K, the
simplest kind of algebraic number field other than the rationals. It has
been proved that there are only finitely many D for which the class
number of K is 1, but it is not known exactly what all these
discriminants are. At least not as far as I know. In any case, this is
certainly a highly non-trivial answer to the original question, albeit
with the conditions relaxed slightly. And it remains within number
theory too...

Andy

PS
The values of D which give class number 1 are: -1,-2,-3,-7,-11,-19,
-43,-67 and -163. These values were conjectured by Gauss to be the only
possibilities. It was 1934 before Heilbronn & Linfoot proved that there
were only finitely many such D. Numerical evidence suggests fairly
strongly that the list above is complete. Good old Gauss...
• ... But of course, pardon me, I was being stupid. Apart from -1, all the numbers in that little list are prime, so it s an ideal answer to the original
Message 7 of 9 , Oct 3, 2003
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> But if looking at finite sets, it does no harm to consider all integers,
> not just primes. And the class number problem provides a good example
> for the original question posed...
> The values of D which give class number 1 are: -1,-2,-3,-7,-11,-19,
> -43,-67 and -163. etc.

But of course, pardon me, I was being stupid. Apart from -1, all the
numbers in that little list are prime, so it's an ideal answer to the
original question.

Andy

(and I don't want anybody trying to say that 1 is prime!)
• In a message dated 03/10/03 15:05:25 GMT Daylight Time, ... You must have an old copy of Hardy & Wright :-) Mine (5th edn.) goes on to say (p.217):- Stark
Message 8 of 9 , Oct 3, 2003
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In a message dated 03/10/03 15:05:25 GMT Daylight Time,
umistphd2003@... writes:

> The values of D which give class number 1 are: -1,-2,-3,-7,-11,-19,
> -43,-67 and -163. These values were conjectured by Gauss to be the only
> possibilities. It was 1934 before Heilbronn & Linfoot proved that there
> were only finitely many such D. Numerical evidence suggests fairly
> strongly that the list above is complete. Good old Gauss...
>

You must have an old copy of Hardy & Wright :-)
Mine (5th edn.) goes on to say (p.217):-
"Stark (1967) proved that this extra field does not exist."

In other words, those are indeed the /only/ simple imaginary quadratic
fields.
I second your salute to the astounding Gauss!

Mike

[Non-text portions of this message have been removed]
• ... Worryingly, I was working from Cohn s advanced number theory, published in 61. That ll teach me to always trust Hardy & Wright more...! Andy
Message 9 of 9 , Oct 3, 2003
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> You must have an old copy of Hardy & Wright :-)
> Mine (5th edn.) goes on to say (p.217):-
> "Stark (1967) proved that this extra field does not exist."

Worryingly, I was working from Cohn's advanced number theory, published
in '61. That'll teach me to always trust Hardy & Wright more...!

Andy
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