Goldbach's conjecture is true for 2n where n is a multiple of 6, anyone disagree?
Proof:
All positive integer multiples of 6 have at least one prime less than n and one prime greater than n that sum to 2n.
ie:
prime1 n prime2
7 54 101
11 54 97
37 54 71
41 54 67
47 54 61
prime1+prime2=2n
7+101=108
11+97=108
37+71=108
41+67=108
47+61=108
n=54 has 5 examples, but there is always at least one of these pairs of primes that sums to 2n, if n is a multiple of 6, meaning that Golbach's conjecture is true for 2n where n is a multiple of 6.
You have a strange idea about what constitutes a proof. Given that Goldbach's conjecture hasn't been disproved, it's hardly surprising that you can find pairs of primes that add up to 2 * 6 * n, where n is less than has been tried in systematic searches for a counter-example to Goldbach's idea.
I'd be very surprised if your narrowed conjecture is true unless Goldbach's conjecture is also true.
A real mathematician might be able to prove that the two conjectures are equivalent.
Multiples of 6 have on average twice as many Goldbach pairs as non-multiples, so if there were a small number of counterexamples it would not be particularly surprising if none of them were multiples of
Of course, it would be very surprising if there were *any* counterexamples!
There was a sci-fi short story I recall reading years ago, where an alien race builds a quantum supercomputer specifically to do "proofs" of that type.
They do it by using exotic matter to construct a "pocket universe" the properties of which they can define, and one of the properties is they set Planck's constant to be as small as possible. They disprove Goldbach's conjecture then by noting that there's a counterexample around the order of 10^43.
They then decide that they want to prove all conjectures of this type with "proofs by inspection", simply by lowering Planck's constant to zero and immediately examining all integers out to infinity. But then bad stuff happens...
IIRC this race didn't really have an economy per-se. Thy had long ago disposed of their mammal bodies and packed all the important guts up into sphere-shaped spacecraft a few meters wide, and lived directly off solar radiation.
It's a series of novels and short stories by the UK author Stephen Baxter. That race is eventually exterminated by humans - all alien races in the galaxy are eventually exterminated by humans as we move colonization outwards from Earth.
For every multiple n of 12 there are multiple sets of two primes (call them p1 and p2) where n-p1 is also a prime, and p1+p2=n (so p2-n is also a prime)
I need to find the sieve rule that proves that then if that is proved then that indirectly proves that Goldbach's conjecture is true for multiples of 12 (and 6).
I'd also be surprised if the number of counter examples were small (but non-zero). The most likely scenarios are that there are either none, or infinitely many, in the latter case, all being large given the searches already made.
A proof that there were a finite number of counter examples would be quite extraordinary.
Not so sure. I think there are other, false, conjectures for which the lowest counterexample is vast. Unfortunately, I've not been able to find a cite.
I know nothing about what a proof of Goldbach's conjecture would be like, so maybe this is all old hat, but...
Has anyone looked for proofs that the number of counter-examples is, as you say, either zero or infinite? I think you were just expressing your intuition there, but intuitions come from somewhere. I know, intuition is not a guarantee, but it might be a sign of a good bet.
I would find it surprising not because it has been verified to to such a large number, but because the number of Goldbach pairs increases so consistently. There aren't any "near misses" known, or even (as far as I know) cases where the number of pairs is even 10% lower than the trend.
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