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

yadda), one particularly obvious yet non-trivial answer to this query would

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.