## The evolution of a tuple

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• Hi all After the symmetrical k +/- 2^n tuple I thought I would try a symmetrical k +/- 2^n*3^m tuple to achieve a somewhat greater density. That is, k +/-
Message 1 of 3 , Jun 2, 2005
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Hi all

After the symmetrical k +/- 2^n tuple I thought I would try a
symmetrical k +/- 2^n*3^m tuple to achieve a somewhat greater density.

That is, k +/- 6,12,18,24,36,48,72,96.... Then the density bug hit
and I thought I would throw in alternating powers of two on different
sides of k, so that i got something like this:

k -24, -18, -16, -12, -6, -4, +2, +6, +8, +12, + 18, +24.

But by then all symmetry was lost. So to heck with symmetry. So I
saw I could sneak in a couple of other numbers as well: 2*11 on one
side and 2*13 on the other, yielding

k -24,-22,-18, -16,-12,-6,-4,+2,+6,+8,+12,+18,+24,+26

And painfully aware that there must be a better way, I took way too
long manually calculating that k must be of the form t*2*3*5*7*11*13
+ 5*7*433.

I thought very cool, that's 14 potential primes in a space of 50,
when there are 'only' 15 primes from 0 to 50 !

Then I started actually looking for the tuple with GP Pari. As it
was working away, I looked on the net about tuples like this. As it
turns out, this particular tuple is well known and in fact is one of
only two minimal 14 tuples. The other tuple has the 2^n * 3^m terms
the same around k, but the power of two terms terms are flipped as
are the 2*11 and 2*13 terms, and k would be a little different, ie,
t*2*3*5*7*11*13 + c, for a c I am not going to calculate this time!

And of course, I have now discovered that guys like Jens have already
found monstrous examples of this 14 tuple, like

26093748*67# + 383123187762431 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32,
36, 42, 48, 50

(33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen. from
http://www.ltkz.demon.co.uk/ktuplets.htm )

Needless to say I then put GP Pari out of its misery trying to see if
this tuple might exist. :)

Anyways, after actually doing the tuple thing, I am now seriously
doubting my belief that the earilest primes will never be surpassed
in density, go figure. :)

Mark
• After doing some reading on the subject I am now firmly convinced that prime constellations with densities exceeding that of the early primes do exist. And
Message 2 of 3 , Jun 2, 2005
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After doing some reading on the subject I am now firmly convinced
that prime constellations with densities exceeding that of the early
primes do exist. And perhaps with quantum computing they shall be
found.

But I am also convinced that the densities will never exceed a
certain amount, as follows:

For instance, there are exactly 168 primes from 1 to 1,000. I
wouldn't be surprised at all now if a prime constellation waaaay up
there somewhere exeeds 168 primes in an interval of 1,000.

*But*, there are exactly 95 primes from 0 to 1000/2. If we include
negative numbers as primes, that makes 190 primes from -500 to 500.

So to my intuition, no interval of 1,000 could ever surpass this 190
count.

Mark

<mark.underwood@s...> wrote:

> Anyways, after actually doing the tuple thing, I am now seriously
> doubting my belief that the earilest primes will never be surpassed
> in density, go figure. :)
>
> Mark
• ... The smallest admissable 168-tuplet is 1033 wide according to an exhaustive search by Thomas J. Engelsma: http://www.opertech.com/primes/k-tuples.html His
Message 3 of 3 , Jun 2, 2005
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Mark Underwood wrote:

> There are exactly 168 primes from 1 to 1,000. I
> wouldn't be surprised at all now if a prime constellation waaaay up
> there somewhere exeeds 168 primes in an interval of 1,000.

The smallest admissable 168-tuplet is 1033 wide according to an exhaustive
search by Thomas J. Engelsma:
http://www.opertech.com/primes/k-tuples.html

His current results say that if there is a prime tuplet closer together than the
earliest primes then it must be at least a 311-tuplet.
From inexhaustive searching, his smallest admissable tuplet beating the earliest
primes is a 447-tuplet of width 3159, where p(447) = 3163. Finding an occurrence
of such a tuplet is out of the question today. However I agree with the k-tuplet
conjecture that there are infinitely many.

--
Jens Kruse Andersen
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