---
id: 5900f5001000cf542c510013
title: 'Problem 403: Lattice points enclosed by parabola and line'
challengeType: 5
forumTopicId: 302071
dashedName: problem-403-lattice-points-enclosed-by-parabola-and-line
---

# --description--

For integers $a$ and $b$, we define $D(a, b)$ as the domain enclosed by the parabola $y = x^2$ and the line $y = ax + b: D(a, b) = \\{ (x, y) | x^2 ≤ y ≤ ax + b \\}$.

$L(a, b)$ is defined as the number of lattice points contained in $D(a, b)$. For example, $L(1, 2) = 8$ and $L(2, -1) = 1$.

We also define $S(N)$ as the sum of $L(a, b)$ for all the pairs ($a$, $b$) such that the area of $D(a, b)$ is a rational number and $|a|,|b| ≤ N$.

We can verify that $S(5) = 344$ and $S(100) = 26\\,709\\,528$.

Find $S({10}^{12})$. Give your answer $\bmod {10}^8$.

# --hints--

`latticePoints()` should return `18224771`.

```js
assert.strictEqual(latticePoints(), 18224771);
```

# --seed--

## --seed-contents--

```js
function latticePoints() {

  return true;
}

latticePoints();
```

# --solutions--

```js
// solution required
```