Goldbach

Goldbach’s conjecture, if we are to believe Wikipedia, asserts that

«Every even integer greater than 2 can be expressed as the sum of two primes.»

And, we are told, this conjecture although widely held to be true, has resisted

all attemps at proof.

I, for one, have a great deal of difficulty with the notion that mathematicians can

come up with mathematical truths - which for reasons known only to the Gods - they

cannot prove.

Indeed, on this one, it is Goldbach’s correspondent, Euler, who agreed it was

undoubtedly true, but couldn’t imagine how to prove it.

We have a situation, here.

What might have been Goldbach’s intent in making such a statement? It seems

to me the statement might be the result of a reflection on larger questions, and our declaration 

one that is the «best that one can say about the matter». Surely

he could not have wanted us completely without parameters in reasoning about

this question.

If every even integer (1) can be seen as the sum of two primes (2) then there is

no one to one representation here. Up to a certain number - «unter einer gegebener Grösse» 

to quote Riemann - odd numbers would have to each serve twice and primes, or some primes,

more than that. Not to be completely lost at sea, what happens if we begin by merely doubling 

the sequence of known primes :

Thus, we have our even integers up to 38 and 3 has served 3 times, 5 has

served 4, and 7 has served 5...and so on. That would seem to me a more germane, 

if modest, statement of Goldbach.