On 21 Mar 00, at 15:59, Vincent J. Mooney Jr. wrote:
> At the new gigahertz speed, it would not take that long to check out all
> even numbers, would it? :-) :-)
_All_ even numbers?
However big an even number is, I can always make a bigger one by
adding 2. Or doubling it. A negative search for a counterexample is
as bad a "proof" as my casual observation that every not-pink object
I can see at the moment happens not to be an elephant, therefore all
elephants are pink ;*)
A proof of Goldbach's Conjecture would surely have to use induction
in some way.
>
> Or find an exception less than 10^100 or so.
Suppose we want to check whether 10^100 conforms to Goldbach's
Conjecture. We would really need a list of all primes up to (very
very nearly) 10^100 to do that. That's a _big_ list.
What I mean is that 10^100 = 3 + (10^100 - 3) is one possible
breakdown. But I don't know offhand whether (10^100 - 3) is prime,
and proving that it is might take some considerable time. If that
fails, I can skip 5 + (10^100 - 5) and 7 + (10^100 - 7) since in each
case the larger number is clearly composite. But, in general, I'd
have to check whether (10^100 - p) is prime for every odd prime p
until I find a prime (proving that Goldbach's Conjecture holds for
the specific odd number 10^100) or until p > 0.5 * 10^100 (a
counterexample!)
Even with terahertz processors, a brute force search up to a "small"
limit like 10^100 would take a very considerable amount of time.
There are no counterexamples up to a fairly large number and,
heuristically, one would expect that it would be decreasingly likely
that increasingly large even numbers could not be decomposed as the
sum of two primes, since the number of decompositions increases
faster than the density of suitable primes decreases. So it looks as
though Goldbach is very probably right. But whether this is provable,
or whether Goldbach's Conjecture is a(nother) victim of Godel's
Theorem, is not (to the best of my knowledge) known.
Regards
Brian Beesley
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