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It is widely believed, that Goldbach's conjecture is true. But suppose, there is a counterexample of, lets say, 50 digits. Is there any hope to prove this counterexample to be one ? Brute force surely would not work. Any ideas ?

Gottfried Helms
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Peter
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  • FYI : http://en.wikipedia.org/wiki/Goldbach%27s_conjecture – mathlove Dec 20 '13 at 10:49
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    If it could have been done, it would have been done. – Torsten Hĕrculĕ Cärlemän Dec 20 '13 at 10:54
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    @Torsten: I don't agree: suppose you could check a single 50-digit candidate in one week of computation. This is not useful for mass searching of all integers in that range, so noone would do that. However, this doesn't automatically imply that no such check can exist and would be tried if someone had good cause to suspect a certain counterexample. That suspicion, on the other hand, would likely have some numerical justification, which also might lead to a more feasible proof than brute force. So it depends on where the supposed counterexample would come from. – MvG Dec 20 '13 at 11:49
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    No. Because it's true. :-) Dirichlet's theorem does NOT necessarily imply, for instance, that ALL numbers are the half-difference of two primes. NOR would a proof of such a statement automatically imply that, by letting the second (subtracted) prime be negative, Goldbach's conjecture is true. But... Well, anyway, what I'm trying to say here is that, IF you want to ‘waste’ you time, effort, and energy on something, just make sure that that certain something has some ‘good chances’ of ultimately being true. – Lucian Dec 20 '13 at 14:20
  • @Lucian Exactly what I meant. Sorry if my sentence went in the wrong direction. – Torsten Hĕrculĕ Cärlemän Dec 20 '13 at 15:41

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The brute force method is effective in this case. Suppose I claim 37998938 is a counterexample to Goldbach. As it happens, the lowest prime that splits 37998938 is 1039. This is the smallest Goldbach-unfriendly number for all primes under 1000. 3325581707333960528 needs the prime 9781 for a Goldbach split.

What is the smallest Goldbach-unfriendly number for all primes under a million? That would be very hard to find, but very easy to check.

For the hypothetical 50-digit counterexample, it would take a few minutes to brute-force verify that the number was an amazing find, working for all primes under a billion. From there, portions would be farmed out for more brute force checking. If the counterexample survived the first hours of incomplete checking, it would become a highly studied number, and would not be ignored.

So far, though, using brute force with primes under 10000 has worked in all cases. Finding an example requiring a prime over 10000 would be a publishable accomplishment.

Ed Pegg
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