From: "Eric Jablow" <erjablow at netacc.net>
To: "WSFA members" <WSFAlist at keithlynch.net>
Subject: [WSFA] Re: Goldbach's Conjecture
Date: Wed, 19 Jun 2002 22:26:19 -0500
Reply-To: WSFA members <WSFAlist at keithlynch.net>

> If mathematicians have been working on this without success since 1742,
> it seems unlikely that a valid proof would be compact enough to fit on 11
> pages.  Mathematicians have presumably also been working since 1742 to
> disprove the conjecture, including using fast computers to look for an
> arithmetic counterexample.
>
> Ron Kean

First, a slight inaccuracy in what I had originally wrote.  In an
algebraic number domain [the ring of algebraic integers in an
algebraic number field, if you must know], a number p is
irreducible if p = ab implies a is a unit or b is a unit.  [You
can't restrict yourself to 'positive numbers' when the ring is
complex.]

Actually, mathematicans probably don't use fast computers to look
for counterexamples to this.  The point of diminishing returns
has been reached.  Also, the conjecture really isn't in the main
line of mathematics these days.  It also doesn't really lead to
anything.

Actually, the Wiles-Taylor Theorem [the Fermat Conjecture, that
is] doesn't lead to any interesting results in and of itself.
Its value is in the various techniques and areas of research that
were discovered by the people trying to prove the conjecture.
Kummer [who originally worked on cyclotomic fields, that is, the
fields Q(\sqrt[n]{1}) ] essentially created the theory of rings
and ideals. We don't use Kummer's notation--we use
Dedekind's--but then we don't use Newton's notation for calculus
either.

Kummer's work led to algebraic number theory, and then to the
beginnings of modern algebraic geometry.  This led to the study
Weil Conjectures, Faltings' proof of the Mordell Conjecture, and
to the important [and don't ask me to explain it here]
Shimura-Taniyama-Weil Conjecture that all elliptic curves are
modular.  Frey showed that the STW Conjecture implied the Fermat
Conjecture.  Wiles showed that STW was true for semi-stable
elliptic curves, and that this was enough for fermat.  Meanwhile,
the full STW was proved two years ago.

Goldbach's Conjecture doesn't lead to much.  Really, there's
every reason to believe that were you to replace the set of
primes with any other set that looks like it statistically, you'd
have at most finitely-many exceptions.

There is a concept in number theory called density.  The
arithmetic density of a set of integers S is the limit as n goes
to infinity of #{ x in S | x <= n}/n.  So, the even numbers have
arithmetic density 1/2, same as the odd numbers.  Primes have
arithmetic density 0.  There are infinitely-many primes, but they
get scarcer and scarcer as one travels out toward infinity.  But,
they do so very slowly.

Remember the prime number theorem: pi(n) = #{x <= n | x prime}
satisfies

  lim pi (n) / (n / ln n) = 1 as n goes to infinity.

In other words, the 'probability' of a random number between 1
and n being prime is about 1/ln n.  This goes to 0 as n goes to
infinity, but so slowly you'd hardly notice.

This leads to the concept of logarithimic density; the log
density of S is the limit as n goes to infinity of

  ln #{x in S | x <= n} / ln n.

The primes have log density 1.  Now, for any set of integers A
and B, define A+B as { a+b | a in A, b in B }.  Log densities
tend to add; the log density of A + B typically is the sum of the
log densities of A and of B.  However, it can't get over 1.  And,
if the sum of those densities exceeds 1, the set tends to be
'full'.

The primes are a 'nice' set in many ways.  They are distributed
fairly regularly.  The density of primes of form 4n+1 equals the
density of primes of form 4n+3, and every similar statement is
true [Dirchlet's Theorem].  Any set like the primes should
satisfy Goldbach with at most finitely many exceptions, and
finitely-many exceptions just aren't interesting.

But, it doesn't lead to anything. No interesting questions depend
on Goldbach.

Respectfully,
Eric Jablow