Aroused by an essay in N.Y. Times of end April 2000 I was inspired to the following arrangement :
Put in a horizontal row the odd primes in their natural order, the same in a vertical column and put at the points of intersection of coordinates the sum of the two primes concerned.
In the main diagonal we will find the double of every prime in natural order, dividing between two mirrored parts; say, in the lower, left part, we find all the possible sums, and we count their appearances according to order.
The total of all splittings up to some limit is equal to the area concerned, and therefore the average of representations of every even sum will grow indefinitely as the number of primes, specially here from i to i6, but naturally there are fluctuations between local maxima and local minima. In the domain under examination (up to 560) these fluctuations stay between twice and half the average about x1.6 and x0.6; statistically there is no tendency for the minimum to be 0, contrary to Goldbach's conjecture.
The question is, if this statistical fact can be extended to a mathematical proof.
My knowledge of this matter goes only up to Vinegradow - who in ca 1947 proved, that above big numbers, 3 primes are sufficient for splitting, meaning, I suppose, that either every even number splits in two odd primes, or the preceding even number will do so.
This fits rather well with the above statistical situation.
But in over fifty years there evidently were trials to bridge this, only apparently small, gap?Herman Baer.
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