## 13177Re: Prime Number Progresions

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• Aug 5 9:05 AM
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Thanks for the correction Jack, I would not have discerned that
difference from what I read unless it was pointed out.

I remember the Hardy-Littlewood k tuple Conjecture but I never did
connect it to polynominals.

Whether the k tuple conjucture is true, I guess I believe it with a
condition, which is that the tuple does not occur if it does not
occur early on, or something like that...

Mark

--- In primenumbers@yahoogroups.com, "jbrennen" <jack@b...> wrote:
> --- Mark Underwood wrote:
>
> > Perhaps 2x^2 + 29 generates the longest sequence of consecutive
> > primes of any two term equation.
>
> If you believe the first Hardy-Littlewood Conjecture (also known
> as the k-tuple Conjecture), there exist arbitrarily long sequences
> of primes from two term equations.
>
> > But for expressions of the form x^2 + x + p there is no need to
> > look for a longer one since it has been shown that p = 41
> > generates the longest one.
>
> Again, the same conjecture implies that arbitrarily long sequences
> of primes exist of the form x^2+x+p.
>
> What has been shown is that p=41 is the largest prime such that
> x^2+x+p is prime for all x, 0 <= x <= p-2.
>
> It has not been shown that x^2+x+p is never prime for 0 <= x <= 40.
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