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.