## 13182Re: Prime Number Progressions

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• Aug 5, 2003
wrote:
>
> > >
> > > > 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.
>
> Surely all one has to do is find a c in the equation x^2+x+c for
> which the following conditions apply:
>
> 2+c not divisible by 2, 3, 5
> and, 2+c meets all of:
>
> 1,5,6 mod 7
> 3,6,8,9,10 mod 11
> 1,3,4,5,6,7,10 mod 13
> 1,3,4,5,8,9,10,12 mod 17
> 4,5,8,11,12,13,14,16,18 mod 19
> 1,3,9,10,11,12,14,16,17,20,21 mod 23
> 3,5,6,7,9,10,12,13,14,16,21,22,26,27 mod 29
> 4,7,11,12,14,15,17,18,19,20,24,26,28,29,30 mod 31
> 1,6,7,8,10,11,12,13,15,16,17,22,24,25,28,32,35,36 mod 37
> 3,4,5,6,7,9,11,14,16,18,19,20,21,22,26,27,30,36,39,40 mod 41
> 1,5,6,8,10,11,14,17,19,22,23,24,26,27,28,29,30,34,36,37,38 mod 43
>
> c=41 is the first number to reach all of the conditions except the
> last, being 1mod7, 10mod11, 4mod13....but 0mod43
>
> Regards
>
> Robert Smith
>
> PS I may have gotten some of the register above incorrect, but
> someone will spot an error if I have made one. Thats what I like