--description--

We call the convex area enclosed by two circles a lenticular hole if:

• The centers 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:

$$\begin{align} & C_0: x^2 + y^2 = 25 \\ & C_1: {(x + 4)}^2 + {(y - 4)}^2 = 1 \\ & C_2: {(x - 12)}^2 + {(y - 4)}^2 = 65 \end{align}$$

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 ($5$, $\sqrt{65}$) 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 $L(100\,000)$.

--hints--

lenticularHoles() should return 4884650818.

assert.strictEqual(lenticularHoles(), 4884650818);

--seed--

--seed-contents--

function lenticularHoles() {

  return true;
}

lenticularHoles();

--solutions--

// solution required