Aug. 30, 1999
Initiation of this directory, "Goldbach_Programs".
This is just for some recreational mathematics stuff that
may be of occasional interest.
I recently read the novel "Uncle Petros and the Goldbach
Conjecture". In the story Petros, but at a time many years in
the past, wonders about whether or not, in particular, the
number 2^100 satisfies GB (so that it is a sum of 2 primes).
Nowadays it is possible to compute answers to questions
of this sort for numbers of that size fairly easily.
As I read the novel and thought about that specific
question I remembered that quite a few years ago, just while
doing recreational work/play with numbers, I had developed a
moderately efficient program to search for the next prime
larger than a given odd number. And I realized that this
program, which I had on file as a MATHEMATICA program, could
be applied to the problem challenge of checking out 2^100
in relation to the Goldbach Conjecture.
The "Strong Goldbach Conjecture"
Noticing that with larger even numbers it seemed to become
possible to find Goldbach decompositions with the smaller prime
very very much smaller than the larger I thought of the simple
conjecture that only a finite set of even integers would be
such that they could not be expressed in the form of a sum of
two primes where the size of the cube of the smaller prime would
be less than the size of the larger prime.
When this was studied computationally (using the special
programs gb2r[x,y,z] and sgbr[x,y]) it turned out that the largest
exception to the rule was larger than it initially seemed that it
might be. Eventually the computing appeared to show that
63274 = 293 + 62681 was the largest even integer exceptional
to the rule. And this was ultimately checked out
computationally for all even numbers out to 1,000,000.
The Programs Checking Primeness
These are not strictly infallible. In particular the
program pta[x] returns "passes" for x=2047 but that is an
error since 2047=23*89. pta[x] is a "strong pseudoprime test"
and ptb[x,n] is a combination of n such tests. This is less
fallible, if properly used.