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Subject: "Prime triples"     Previous Topic | Next Topic
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neat_maths
Member since Aug-22-03
Jul-10-09, 11:19 PM (EST)
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"Prime triples"
 
   It is conjectured, but not proved that prime number pairs that differ by 2 will exist ad infinitum.

Is there another prime triple higher than 3, 5, 7 where the sequence differs by 2 then 2 ?

take care


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alexbadmin
Charter Member
2410 posts
Jul-10-09, 11:24 PM (EST)
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2. "RE: Prime triples"
In response to message #0
 
   3, 5, 7 is the only such triple because of the three numbers n, n ± 2 one is always divisible by 3.


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neat_maths
Member since Aug-22-03
Jul-12-09, 02:09 PM (EST)
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3. "Prime triples"
In response to message #2
 
   Similarly is 3, 7, 11 the only prime triple which is 4 apart because one of (n minus 4), n, (n plus 4) must be divisible by 3 ?

Also is 3, 11, 19 the only prime triple that is 8 apart for the same reason ?

Is there any prime triple which is 16 apart ?

Is there any prime triple which is 32 apart ?

Is there any prime triple which is 64 apart above 3, 67, 131 ?

Is there any prime triple which is 128 apart ? 256 ? 512 ? 1024 ?


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alexbadmin
Charter Member
2410 posts
Jul-12-09, 02:16 PM (EST)
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4. "RE: Prime triples"
In response to message #3
 
   >Similarly is 3, 7, 11 the only prime triple which is 4 apart
>because one of (n minus 4), n, (n plus 4) must be divisible
>by 3 ?

That is correct.

>
>Also is 3, 11, 19 the only prime triple that is 8 apart for
>the same reason ?

Yes, this is also correct.

>Is there any prime triple which is 16 apart ?

2k = ± 1 (mod 3) for any k > 0.

Therefore all such triples (and the ones below) contain a number divisible by 3. But the only prime number divisible by 3 is 3 itself. It follows that if n and n ± 2k are all prime then necessarily n - 2k = 3.

Since 3 + 16 = 19, 19 + 16 = 35, there is no three term arthmetic progression with the difference 16 in which all three numbers are prime.


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alexbadmin
Charter Member
2410 posts
Jul-12-09, 02:18 PM (EST)
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5. "RE: Prime triples"
In response to message #4
 
   Of course if negative numbers are allowed then we have triples

-5, 3, 11
-13, 3, 19
-61, 3, 67
...


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