a9418a1fe9
* fix: clean-up Project Euler 221-240 * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
63 lines
1.4 KiB
Markdown
63 lines
1.4 KiB
Markdown
---
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id: 5900f4521000cf542c50ff64
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title: 'Problem 229: Four Representations using Squares'
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challengeType: 5
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forumTopicId: 301872
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dashedName: problem-229-four-representations-using-squares
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---
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# --description--
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Consider the number 3600. It is very special, because
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$$\begin{align}
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& 3600 = {48}^2 + {36}^2 \\\\
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& 3600 = {20}^2 + {2×40}^2 \\\\
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& 3600 = {30}^2 + {3×30}^2 \\\\
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& 3600 = {45}^2 + {7×15}^2 \\\\
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\end{align}$$
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Similarly, we find that $88201 = {99}^2 + {280}^2 = {287}^2 + 2 × {54}^2 = {283}^2 + 3 × {52}^2 = {197}^2 + 7 × {84}^2$.
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In 1747, Euler proved which numbers are representable as a sum of two squares. We are interested in the numbers $n$ which admit representations of all of the following four types:
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$$\begin{align}
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& n = {a_1}^2 + {b_1}^2 \\\\
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& n = {a_2}^2 + 2{b_2}^2 \\\\
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& n = {a_3}^2 + 3{b_3}^2 \\\\
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& n = {a_7}^2 + 7{b_7}^2 \\\\
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\end{align}$$
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where the $a_k$ and $b_k$ are positive integers.
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There are 75373 such numbers that do not exceed ${10}^7$.
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How many such numbers are there that do not exceed $2 × {10}^9$?
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# --hints--
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`representationsUsingSquares()` should return `11325263`.
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```js
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assert.strictEqual(representationsUsingSquares(), 11325263);
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```
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# --seed--
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## --seed-contents--
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```js
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function representationsUsingSquares() {
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return true;
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}
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representationsUsingSquares();
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```
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# --solutions--
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```js
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// solution required
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```
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