9109

This number is a prime.

Mark n points on a circle and connect every pair of points with a line, then count the number of regions (r) into which the interior of the circle is divided. It is a theorem that if the points on the circle are chosen such that no three lines intersect at the same point inside the circle, the total number of regions will be maximized, so assume this condition holds. As an example, the diagram below shows that when n=5 there are r=16 regions. Is it possible for both n and r to be prime? Yes, and aside from the somewhat trivial case n=2 and r=2 the smallest such configuration has 23 points and 9109 regions. The next one after that is n=73, r=1091059.

[Keith]