freeCodeCamp/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-318-2011-nines.md

---
id: 5900f4ab1000cf542c50ffbd
title: 'Problem 318: 2011 nines'
challengeType: 5
forumTopicId: 301974
dashedName: problem-318-2011-nines
---

# --description--

Consider the real number √2+√3.

When we calculate the even powers of √2+√3

we get:

(√2+√3)2 = 9.898979485566356...

(√2+√3)4 = 97.98979485566356...

(√2+√3)6 = 969.998969071069263...

(√2+√3)8 = 9601.99989585502907...

(√2+√3)10 = 95049.999989479221...

(√2+√3)12 = 940897.9999989371855...

(√2+√3)14 = 9313929.99999989263...

(√2+√3)16 = 92198401.99999998915...

It looks like that the number of consecutive nines at the beginning of the fractional part of these powers is non-decreasing. In fact it can be proven that the fractional part of (√2+√3)2n approaches 1 for large n.

Consider all real numbers of the form √p+√q with p and q positive integers and p&lt;q, such that the fractional part of (√p+√q)2n approaches 1 for large n.

Let C(p,q,n) be the number of consecutive nines at the beginning of the fractional part of (√p+√q)2n.

Let N(p,q) be the minimal value of n such that C(p,q,n) ≥ 2011.

Find ∑N(p,q) for p+q ≤ 2011.

# --hints--

`euler318()` should return 709313889.

```js
assert.strictEqual(euler318(), 709313889);
```

# --seed--

## --seed-contents--

```js
function euler318() {

  return true;
}

euler318();
```

# --solutions--

```js
// solution required
```
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`---`
			`id: 5900f4ab1000cf542c50ffbd`
			`title: 'Problem 318: 2011 nines'`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`challengeType: 5`
fix(curriculum): Added forumTopicId to remaining 1200 challeng… (#36558) 2019-08-05 16:17:33 +00:00			`forumTopicId: 301974`
feat(curriculum): restore seed + solution to Chinese (#40683) * feat(tools): add seed/solution restore script * chore(curriculum): remove empty sections' markers * chore(curriculum): add seed + solution to Chinese * chore: remove old formatter * fix: update getChallenges parse translated challenges separately, without reference to the source * chore(curriculum): add dashedName to English * chore(curriculum): add dashedName to Chinese * refactor: remove unused challenge property 'name' * fix: relax dashedName requirement * fix: stray tag Remove stray `pre` tag from challenge file. Signed-off-by: nhcarrigan <nhcarrigan@gmail.com> Co-authored-by: nhcarrigan <nhcarrigan@gmail.com> 2021-01-13 02:31:00 +00:00			`dashedName: problem-318-2011-nines`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`---`

Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`# --description--`

feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`Consider the real number √2+√3.`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`When we calculate the even powers of √2+√3`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`we get:`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`(√2+√3)2 = 9.898979485566356...`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`(√2+√3)4 = 97.98979485566356...`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`(√2+√3)6 = 969.998969071069263...`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`(√2+√3)8 = 9601.99989585502907...`
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`(√2+√3)10 = 95049.999989479221...`

Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`(√2+√3)12 = 940897.9999989371855...`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`(√2+√3)14 = 9313929.99999989263...`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`(√2+√3)16 = 92198401.99999998915...`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`It looks like that the number of consecutive nines at the beginning of the fractional part of these powers is non-decreasing. In fact it can be proven that the fractional part of (√2+√3)2n approaches 1 for large n.`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`Consider all real numbers of the form √p+√q with p and q positive integers and p<q, such that the fractional part of (√p+√q)2n approaches 1 for large n.`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`Let C(p,q,n) be the number of consecutive nines at the beginning of the fractional part of (√p+√q)2n.`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
			`Let N(p,q) be the minimal value of n such that C(p,q,n) ≥ 2011.`

			`Find ∑N(p,q) for p+q ≤ 2011.`

Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`# --hints--`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`euler318()` should return 709313889.
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			```js
			`assert.strictEqual(euler318(), 709313889);`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			```

Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`# --seed--`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`## --seed-contents--`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
			```js
			`function euler318() {`
fix: remove Good luck! comment from challenge seed code (#39575) 2020-09-15 16:57:40 +00:00
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00			`return true;`
			`}`

			`euler318();`
			```

Feat: add new Markdown parser (#39800) and change all the challenges to new `md` format. 2020-11-27 18:02:05 +00:00			`# --solutions--`
feat(challenge-md): Add initial markdown challenge files 2018-09-30 22:01:58 +00:00
			```js
			`// solution required`
			```