Question concerning A092749.

[Pieter Moree]

Robert G. Wilson wrote:

> Et al,
>
> 	I believe that this is simply Euler's "Lucky" numbers: n such that
> m^2-m+n is
> prime for m=0..n-1,
> http://www.research.att.com/projects/OEIS?Anum=A014556 .
>
> Bob.
>
> %I A092749
> %S A092749
> 2,3,5,5,11,11,11,11,11,11,17,17,17,17,17,17,41,41,41,41,41,41,41,41,41,
> %T A092749 41,41,41,41,41,41,41,41,41,41,41,41,41,41,41
> %N A092749 a(n) = least k such that m^2 + m + k is prime for m = 0, 1,
> ... n-1. %e A092749 a(2) = 3 because 0^2 + 0 + 3 = 3 is prime, and 1^2
> + 1 + 3 = 5 is  prime, and it is the smallest number with the required
> properties. %K A092749 more,nonn,new
> %O A092749 1,1
> %A A092749 Gabriel Cunningham (gcasey(AT)mit.edu), Apr 12 2004

The numbers 2,3,5,11,17 and 41 above are the only numbers A
such that m^2+m+A is prime for m=0,....,A-2 (this can be proved,
see Mollin's paper and is closely related to the celebrated
Rabinowitsch criterium).

Since the value of m^2+m+A is A^2 for m=A-1, one cannot possible
do better than this.

An obvious question of course is whether for given n, a(n) exists
at all. This is far from obvious. Assuming the generally believed
k-tuplets conjecture the answer is yes as was shown by
Andrew Granville. For a proof (which is not
very difficult) see the paper:

Mollin, R. A. Prime-producing quadratics.  Amer. Math. Monthly  104
(1997),  no. 6, 529--544.

It is also known, due to work of

Lukes, Patterson and Williams
(Numerical sieving devices: their history and some applications,
Nieuw Archief Wisk. 13 (1995), 113-139)

that any further elements
in the above sequence, if they exist, are >10^{18} !

So don't let your computer run on this....

Unfortunately it seems that for the computer adepts the area of
prime producing polynomials is well-grazed. Some progress was
made by Hugh Williams et al. using special purpose computers.
This was about find polynomials of the form x^2+x+A that
asymptotically represent many primes.

For example Beeger's polynomial m^2+m+27941 gives up to 10^6,
286128 primes, whereas Euler's (with 41) gives 261080.

Presently I am writing a note on  a closely related problem which in
contrast seems to be completely unexplored.

I had much fun doing computer experiments there setting new records. I
am sure though that it is a matter of putting in more time to
do better there. Shortly I hope to put my note on this on the
ArXiv. It presents the record-hunting interested people with
some challenges.

Pieter Moree