Minimal number of different distances

[Dan Dima]

Frank can you share both your distributions for a(19) <= 8 & a(12) <= 5
(maybe some graphics will help). I can't figure either if there is a smaller
n such that a(n) < floor(n/2).
At the first sight it seems to me that hexagonal grids with n points are not
better than the regular n-gon.
As I have seen on http://schoolar.google.com Erdos conjectured that a(n) =
floor(n/2).
Sorry for using non-standard notations, I will follow the rules next time.

Regards,
Dan


On 5/3/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
>   I will note that a_2(n) is not always floor(n/2); in particular, 19
> points in a regular hexagonal pattern have only 8 distances and 12 points in
> a slightly irregular hexagon have only 5 distances.  I think hexagonal grids
> give a bound of n/4 + o(n) on a_2(n).  (It may be lower, since there are
> non-trivial solutions to a^2+ab+b^2 = c^2+cd+d^2; I'm not immediately able
> to estimate their density.)
>
> Franklin T. Adams-Watters
>
>
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