1.6 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f5461000cf542c510058 | Problem 473: Phigital number base | 5 | 302150 | problem-473-phigital-number-base |
--description--
Let \\varphi be the golden ratio: \\varphi=\\frac{1+\\sqrt{5}}{2}.
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.
Even then this representation is not unique.
We can make it unique by requiring that no powers with consecutive exponents are used and that the representation is finite.
E.g:
2=\\varphi+\\varphi^{-2} and 3=\\varphi^{2}+\\varphi^{-2}
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).
The sum of the positive integers not exceeding 1000 whose phigital representation is palindromic is 4345.
Find the sum of the positive integers not exceeding 10^{10} whose phigital representation is palindromic.
--hints--
euler473() should return 35856681704365.
assert.strictEqual(euler473(), 35856681704365);
--seed--
--seed-contents--
function euler473() {
return true;
}
euler473();
--solutions--
// solution required