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   Author  Topic: Goldbach grids  (Read 1695 times)
JocK
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Goldbach grids  
« on: Jun 26th, 2005, 6:39am »
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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?
 
 
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xy - y = x5 - y4 - y3 = 20; x>0, y>0.
towr
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Re: Goldbach grids  
« Reply #1 on: Jun 26th, 2005, 7:40am »
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Is it infinite in all direction, or just two?
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JocK
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Re: Goldbach grids  
« Reply #2 on: Jun 26th, 2005, 9:26am »
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on Jun 26th, 2005, 7:40am, towr wrote:
Is it infinite in all direction, or just two?

 
It's an in finite square grid without edges.
 
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.

xy - y = x5 - y4 - y3 = 20; x>0, y>0.
Barukh
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Re: Goldbach grids  
« Reply #3 on: Jun 26th, 2005, 10:54am »
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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

 
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Re: Goldbach grids  
« Reply #4 on: Jun 26th, 2005, 11:16am »
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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

« Last Edit: Jun 26th, 2005, 11:26am by towr » IP Logged

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JocK
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Re: Goldbach grids  
« Reply #5 on: Jun 26th, 2005, 11:46am »
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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.
 
 
 
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... Grin
 
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?

 
« Last Edit: Jun 26th, 2005, 12:30pm by JocK » IP Logged

solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.

xy - y = x5 - y4 - y3 = 20; x>0, y>0.
Ajax
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Re: Goldbach grids  
« Reply #6 on: Jun 27th, 2005, 11:19pm »
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A question: What's the difference between prime numbers and odd prime numbers?
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Barukh
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Re: Goldbach grids  
« Reply #7 on: Jun 28th, 2005, 1:00am »
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on Jun 27th, 2005, 11:19pm, 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.
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