55 lines
1.6 KiB
Markdown
55 lines
1.6 KiB
Markdown
---
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id: 5900f5461000cf542c510058
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title: 'Problem 473: Phigital number base'
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challengeType: 5
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forumTopicId: 302150
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dashedName: problem-473-phigital-number-base
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---
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# --description--
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Let $\\varphi$ be the golden ratio: $\\varphi=\\frac{1+\\sqrt{5}}{2}.$
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Remarkably it is possible to write every positive integer as a sum of powers of $\\varphi$ even if we require that every power of $\\varphi$ is used at most once in this sum.
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Even then this representation is not unique.
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We can make it unique by requiring that no powers with consecutive exponents are used and that the representation is finite.
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E.g:
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$2=\\varphi+\\varphi^{-2}$ and $3=\\varphi^{2}+\\varphi^{-2}$
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To represent this sum of powers of $\\varphi$ we use a string of 0's and 1's with a point to indicate where the negative exponents start. We call this the representation in the phigital numberbase. So $1=1*{\\varphi}$, $2=10.01*{\\varphi}$, $3=100.01*{\\varphi}$ and $14=100100.001001*{\\varphi}$. The strings representing 1, 2 and 14 in the phigital number base are palindromic, while the string representing 3 is not. (the phigital point is not the middle character).
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The sum of the positive integers not exceeding 1000 whose phigital representation is palindromic is 4345.
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Find the sum of the positive integers not exceeding $10^{10}$ whose phigital representation is palindromic.
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# --hints--
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`euler473()` should return 35856681704365.
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```js
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assert.strictEqual(euler473(), 35856681704365);
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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 euler473() {
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return true;
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}
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euler473();
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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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