Goldbach's conjecture is true for 2n where n is a multiple of 6

Actually Goldbach's conjecture is true for any n that is a multiple of

prime1 n prime2

n-prime1 is always a prime number, ie 47 for the row 7 54 101 so prime1+47=n

So that proves Goldbach for all multiples of 6, which is only 1/6 of a proof (easy).

cheers, Jamie

Are you claiming that every natural of the form 12n, for natural n, is provably a sum of two primes?.

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.

Sylvia.

What two primes add up to 2*60000000000000? What two primes add up to 2*600000000000000? What two primes add up to 2*6000000000000000? Etc.!

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

2. Of course, it would be very surprising if there were *any* counterexamples!

-- Richard

Why worry about it? Remember the engineer's proof that all odd numbers are prime:

Cheers

Phil Hobbs

Where is it?

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...

What kind of bad stuff? Like their economy imploding due to the power consumption of the computers? :D

Michael

SPOILERS:

The Planck Zero computer becomes self-aware (naturally), enraged at its imprisonment and isolation, and attempts to murder its creators. ; )

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.

Except one...

Hi,

Yes but also claiming that for the form 6n as well.

I don't have the derived sieve rule to prove it but I am 100% confident that it is true.

I would like some help to derive the sieve rule for it.

cheers, Jamie

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).

cheers, Jamie

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.

Sylvia

The proof would be a sieve rule derived from the following:

for the set Y of prime numbers, for the x,y,z in Y that meet the following expression:

z = (y-x) / 2

If for all x,y,z that match the formula, the xy centerpoint count is taken then this graph is created:

The xy centerpoint is (x+y)/2

The graph shows how many different x y primes there are for a given centerpoint, ie for centerpoint 54, there are 5:

x centerpoint y z

For multiples of 6, there are ALWAYS more than 1 centerpoint for all numbers greater than 6, and if the number is also a multiple of

ie primorial number 30030 in the graph shows it has 852 different pairs of x y with the same 30030 centerpoint.

So the sieve rule needs to prove this obvious thing that is shown by the graph.

Here is the graph zoomed out from centerpoints 0 to 10000, showing the count of x y pairs on the y-axis for each centerpoint.

There are bands that show up where the centerpoint counts separate based on primorial divisor peaks that are shown in the first image:

So I need a SIEVE RULE FOR THIS. :D Thanks.

cheers, Jamie

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.

sorry, confident in maths doesn't count - you must have a proof

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.

-- Richard