## Re: The evolution of a tuple

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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
Message 1 of 3 , Jun 2, 2005
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 2 of 3 , Jun 2, 2005
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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