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CTK Exchange
neat_maths
Member since Aug-22-03
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Jul-10-09, 11:19 PM (EST) |
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"Prime triples"
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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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neat_maths
Member since Aug-22-03
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Jul-12-09, 02:09 PM (EST) |
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3. "Prime triples"
In response to message #2
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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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alexb
Charter Member
2410 posts |
Jul-12-09, 02:16 PM (EST) |
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4. "RE: Prime triples"
In response to message #3
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>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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