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