[FOM] 570: Philosophy of Incompleteness 1

[Timothy Y. Chow]

Harvey Friedman wrote:

> I'll close here with the following intriguing example: every even 
> element of {1,...,52!} is the sum of two primes. To have this meet 
> physical reality, one has to examine actual primality testing. There are 
> fast testers using Monte Carlo, and there are slower testers that don't. 
> Both use Higher Mathematics. I don't know where the slower testers break 
> down at the practical level - below 52! or below 100! or where? Experts 
> surely can answer this for us. And we still have to tell whether a given 
> even number in the relevant range is in fact the sum of two primes.

The following is taken from Wikipedia.

     http://en.wikipedia.org/wiki/Elliptic_curve_primality

"As of 2011 the largest prime that has been proved with ECPP method is the 
26,643-digits prime value of the Ramanujan tau function:

          LR(157,2207) = tau(157^2206).

The distributed computation with fastECPP software by Francois Morain 
started in January 2011 and ended in April 2011. The total CPU time is 
equal to 2355 hours."

Note, however, that if one wants to computationally verify Goldbach up to 
some bound, then primality proving doesn't really come into it; one uses 
the sieve of Eratosthenes to generate primes.

     http://sweet.ua.pt/tos/goldbach.html

But these techniques will obviously not get up to anywhere near 52!.

Tim