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        riddles >> hard >> Goldbach grids
        
(Message started by: JocK on Jun 26^th, 2005, 6:39am)

Title: Goldbach grids
Post by JocK on Jun 26^th, 2005, 6:39am

We define a Goldbach grid as an infinite square grid filled with odd primes such that each three subsequent primes - either in a row or in a column - add up to a prime number.

The following example shows a Goldbach grid

3 5 3 5 3 5
5 7 5 7 5 7
3 5 3 5 3 5
5 7 5 7 5 7
3 5 3 5 3 5
5 7 5 7 5 7

(constructed by repeating a simple 2x2 pattern) that contains the first three odd primes.

Can you create Goldbach grids containing the first 4, 5, 6, .. odd primes?

Title: Re: Goldbach grids
Post by towr on Jun 26^th, 2005, 7:40am

Is it infinite in all direction, or just two?

Title: Re: Goldbach grids
Post by JocK on Jun 26^th, 2005, 9:26am

on 06/26/05 at 07:40:39, towr wrote:

Is it infinite in all direction, or just two?

It's an in finite square grid without edges.

Title: Re: Goldbach grids
Post by Barukh on Jun 26^th, 2005, 10:54am

Another question: should every prime number appear an infinite number of times? I suspect the answer is yes. Otherwise, the following filling works for the first 4 numbers:

3 5 3 5 3 5 3
5 5 3 5 3 5 5
3 3 7 3 7 3 3
5 5 3 11 3 5 5
3 3 7 3 7 3 3
5 5 3 5 3 5 5
3 5 3 5 3 5 3

Title: Re: Goldbach grids
Post by towr on Jun 26^th, 2005, 11:16am

for 4 (with each prime number an infinite number of times):

5 3 3 7 3 3
11 3 3 11 3 3
7 5 7 5 7 5
5 3 3 7 3 3
11 3 3 11 3 3
7 5 7 5 7 5

Title: Re: Goldbach grids
Post by JocK on Jun 26^th, 2005, 11:46am

A point of clarification:

the idea is to construct a rectangle of NxM primes that can be repeated. When you have found one, it suffices to post the reptitive unit only.

So, a repetitive unit using the first four primes could be:

3 5 3
11 7 11
3 5 3

as it can be used to construct a valid Goldbach grid:

3 5 3 3 5 3
11 7 11 11 7 11
3 5 3 3 5 3
3 5 3 3 5 3
11 7 11 11 7 11
3 5 3 3 5 3

I think this is the easiest way to generate infinite Goldbach grids.

[hide]Of course you can change one 11 in the above repetitive unit into 13. And you can also change one 5 into 17. And you can do both changes together... ;D

In fact, it is not too difficult to find a valid 3x3 reptitive unit containing the first 9 odd primes...

Who is the first to find a Goldbach grid that goes beyond the first 9 odd primes?[/hide]

Title: Re: Goldbach grids
Post by Ajax on Jun 27^th, 2005, 11:19pm

A question: What's the difference between prime numbers and odd prime numbers?

Title: Re: Goldbach grids
Post by Barukh on Jun 28^th, 2005, 1:00am

on 06/27/05 at 23:19:59, Ajax wrote:

A question: What's the difference between prime numbers and odd prime numbers?

2. That is, every prime number except 2 is odd.

Applying this to the problem at hand: Goldbach grid doesn't contain number 2.