## A goldbach conjecture for triplet primes

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• ... Sorry, Phil. I was rushing to get my check ... to reach msg #4208, and forgot that you had said the above in #4164. Conjecture: The largest even integer
Message 1 of 6 , Nov 30, 2001
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Phil reminded us of his question:

> HAve you tried densities similar to those of twin primes,
> (1/ln(n)^2), or higher power ln expressions (all denser
> than the n^(1/a) spacings in the long run, but we don't
> have the CPU power to exhaustively check anything but the
> initial billion or so, which is a small number).

Sorry, Phil. I was rushing to get my check
> 4204,4206,4208]
to reach msg #4208, and forgot that you had said the above in #4164.

Conjecture: The largest even integer that is not
the sum of two triplet primes is 7400384

David
• ... Grin - pedant warning! Which triples? {0,2,6} or {0,4,6}, or either? ... It matters not - 7 million s far too low. I take your conjecture, and raise it
Message 2 of 6 , Nov 30, 2001
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On Fri, 30 November 2001, d.broadhurst@... wrote:
> Conjecture: The largest even integer that is not
> the sum of two triplet primes is 7400384

Grin - pedant warning!
Which triples? {0,2,6} or {0,4,6}, or either?
:-)

It matters not - 7 million's far too low. I take your conjecture, and raise it "only triplet primes not of the form 6n", i.e. the _middles_ alone. (+2+4, +4+4 and +2+2 do cover the three residues mod six, so we're alright.)

No I haven't got a lower bound, but then again I'm sat in front of a pathetic machine which came shipped with _no_ compiler as standard, bizarre! If noone does the donkey-work for me, I am of course prepared to do it myself.

Phil

Don't be fooled, CRC Press are _not_ the good guys.
They've taken Wolfram's money - _don't_ give them yours.
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• Phil Carmody ... I took my defn from Chris Caldwell, so the answer is: either. Defn: A prime triple is three consecutive primes, such that the first and the
Message 3 of 6 , Nov 30, 2001
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Phil Carmody

> Which triples? {0,2,6} or {0,4,6}, or either?

I took my defn from Chris Caldwell, so the answer is: either.

Defn: A prime triple is three consecutive primes,
such that the first and the last differ by six.

Defn: A prime p is a triplet prime if and only if
it belongs to a prime triple.

Conjecture: The largest even integer that is not
the sum of two triplet primes is 7400384

I tried only up to sum=10^8
with ZERO checking :-(

David
• Phil Carmody ... I repeated the calculation leaving out all primes divisible by 6. It apears that the conclusion is unchanged. David:-)
Message 4 of 6 , Dec 1, 2001
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Phil Carmody
> I take your conjecture, and raise it
> "only triplet primes not of the form 6n",
I repeated the calculation leaving out
all primes divisible by 6.
It apears that the conclusion is unchanged.
David:-)
• Phil: As for triplet middles : try the analyis mod 30. David
Message 5 of 6 , Dec 1, 2001
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Phil: As for "triplet middles": try the analyis mod 30. David
• ... As you may have worked out - I did have my doubts, then I had my doubts about my doubts :-|. However, I m convinced that for at least 10 minutes last night
Message 6 of 6 , Dec 1, 2001
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On Sat, 01 December 2001, d.broadhurst@... wrote:

>
> Phil: As for "triplet middles": try the analyis mod 30. David

As you may have worked out - I did have my doubts, then I had my doubts about my doubts :-|. However, I'm convinced that for at least 10 minutes last night I was actualy correct, evem if for the wrong reason.

Explicitly:
1 5 7 5 alone
5 7 11 7 alone
7 11 13 30n+11
11 13 17 30n+13
13 17 19 30n+17
17 19 23 30n+19
Therefore the gamut of sums is
30n + { 0, 4, 6, 8, 22, 24, 26, 28 }
and occasional 30n + { 16, 18, 20 }
Which falls somewhat short of the mark.

Oh well.

Phil

Don't be fooled, CRC Press are _not_ the good guys.
They've taken Wolfram's money - _don't_ give them yours.
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