## Lenticular holes

### Problem 295

Published on Saturday, 5th June 2010, 01:00 pm; Solved by 331; Difficulty rating: 75%We call the convex area enclosed by two circles a *lenticular hole* if:

- The centres of both circles are on lattice points.
- The two circles intersect at two distinct lattice points.
- The interior of the convex area enclosed by both circles does not contain any lattice points.

Consider the circles:

C_{0}: `x`^{2}+`y`^{2}=25

C_{1}: (`x`+4)^{2}+(`y`-4)^{2}=1

C_{2}: (`x`-12)^{2}+(`y`-4)^{2}=65

The circles C_{0}, C_{1} and C_{2} are drawn in the picture below.

C_{0} and C_{1} form a lenticular hole, as well as C_{0} and C_{2}.

We call an ordered pair of positive real numbers (r_{1}, r_{2}) a *lenticular pair* if there exist two circles with radii r_{1} and r_{2} that form a lenticular hole.
We can verify that (1, 5) and (5, √65) are the lenticular pairs of the example above.

Let L(N) be the number of **distinct** lenticular pairs (r_{1}, r_{2}) for which 0 < r_{1} ≤ r_{2} ≤ N.

We can verify that L(10) = 30 and L(100) = 3442.

Find L(100 000).