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Generalised Goldbach Conjecture

Anonymous Wed 27 May 19:50:28 2020 No.11721913 View Reply Original Report

Quoted By: >>11724024 >>11724034 >>11724103

What's the "smallest" set of natural numbers such that an analogous form of the goldbach conjecture holds, meaning that for any positive integer n in N there are a and b in the Set such that a + b = n? Is there a name for such a conjecture or is the answer obvious?

pic not really related

Anonymous

Anonymous Wed 27 May 2020 21:25:47 No.11723887 Report

Quoted By:

bump, as if nobody here is interested in this stuff

Anonymous

Anonymous Wed 27 May 2020 21:46:13 No.11724024 Report

Quoted By: >>11724097

>>11721913
It's not appropriate to call it generalized goldbach because a resolution to it would not necessarily yield a resolution to goldbach conjecture.

Anonymous

Anonymous Wed 27 May 2020 21:47:30 No.11724034 Report

Quoted By: >>11724086

>>11721913
Also what the fuck do you mean by smallest? Any set like that must be infinite and is very likely to have natural density zero, so what the fuck do you mean by a size here?

Anonymous

Anonymous Wed 27 May 2020 21:54:57 No.11724086 Report

Quoted By:

>>11724034
Isn't it obvious when talking about natural numbers? I think a definition following the intuition would be that assuming we have 2 sets, take the difference of the sizes of the subsets of both sets such that all the elements of the subsets are below a certain n. Then as n goes to infinity, you will see whether that difference will be positive or negative, from which you can deduce which one contains "more" integers.

Anonymous

Anonymous Wed 27 May 2020 21:56:13 No.11724097 Report

Quoted By:

>>11724024
Sure, it's just that I know of no other conjecture like it which is why I thought that name to be appropriate.

Anonymous

Anonymous Wed 27 May 2020 21:56:59 No.11724103 Report

Quoted By: >>11724133

>>11721913

im going to guess 2^n.

why? because we know there is always a prime between prime p and 2p. (I think.)

Anonymous

Anonymous Wed 27 May 2020 22:01:36 No.11724133 Report

Quoted By:

>>11724103
No. Your set would be {1, 2, 4, 8, 16, 32... } and you could not write 7 as the sum of two elements of the set.

Anonymouse

Anonymouse Wed 27 May 2020 22:17:18 No.11724231 Report

Quoted By: >>11724283

For any finite set of k generators, you can generate at most k*k/2 distinct sums. So you need at least sqrt(2*n) generators to be less than n. I presume you include 0 as a natural number, otherwise you can't get 1 as a sum of two natural numbers.

Anonymous

Anonymous Wed 27 May 2020 22:27:09 No.11724283 Report

Quoted By: >>11724466

>>11724231
Let's say every natural number bigger than some finite constant (which shouldn't be pushed too far, for example the goldbach conjecture concerns every even integer greater than 2)
Still, that doesn't solve the problem.
Also it appears to me that an iterative approach wouldn't work, as we would just generate the natural numbers which do very obviously not solve the problem.

Anonymouse

Anonymouse Wed 27 May 2020 23:07:15 No.11724466 Report

Quoted By: >>11725748

>>11724283
Here's a pattern that, asymptotically, is off by no mare than a factor of sqrt(2):
0,1,2,3,4,5,6,7,8,9
10,20,30,40,50,60,70,80,90,
100,101,102,...,199,
200, 300, 400, 500, 600, 700, 800, 900,
1000,1001,1002,...,1999,
and so forth It might need some tweaking to work properly.

Anonymous

Anonymous Thu 28 May 2020 07:27:47 No.11725748 Report

Quoted By: >>11725787

Cute problem OP, I wonder if anyone has looked into this before (probably).
By the way, a way you can define "smallest" set is if it doesn't have any proper subsets that satisfy the properties. The downside being there may be a lot of "smallest" sets. >>11724466 might be one.

Anonymous

Anonymous Thu 28 May 2020 07:55:20 No.11725787 Report

Quoted By:

>>11725748
But then the problem is trivial. The anon you quoted already found a minimal set.

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