---
id: 5900f3ae1000cf542c50fec1
title: 'Problem 66: Diophantine equation'
challengeType: 5
forumTopicId: 302178
dashedName: problem-66-diophantine-equation
---

# --description--

Consider quadratic Diophantine equations of the form:

<div style='text-align: center;'>x<sup>2</sup> – Dy<sup>2</sup> = 1</div>

For example, when D=13, the minimal solution in x is 649<sup>2</sup> – 13×180<sup>2</sup> = 1.

It can be assumed that there are no solutions in positive integers when D is square.

By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following:

<div style='margin-left: 2em;'>
  3<sup>2</sup> – 2×2<sup>2</sup> = 1<br>
  2<sup>2</sup> – 3×1<sup>2</sup> = 1<br>
  <strong><span style='color: red;'>9</span></strong><sup>2</sup> – 5×4<sup>2</sup> = 1<br>
  5<sup>2</sup> – 6×2<sup>2</sup> = 1<br>
  8<sup>2</sup> – 7×3<sup>2</sup> = 1<br>
</div>

Hence, by considering minimal solutions in `x` for D ≤ 7, the largest `x` is obtained when D=5.

Find the value of D ≤ 1000 in minimal solutions of `x` for which the largest value of `x` is obtained.

# --hints--

`diophantineEquation()` should return a number.

```js
assert(typeof diophantineEquation() === 'number');
```

`diophantineEquation()` should return 661.

```js
assert.strictEqual(diophantineEquation(), 661);
```

# --seed--

## --seed-contents--

```js
function diophantineEquation() {

  return true;
}

diophantineEquation();
```

# --solutions--

```js
// solution required
```