Irrational Base

Let $r$ be the real root of the equation $x^3 = x^2 + 1$.
Every positive integer can be written as the sum of distinct increasing powers of $r$.
If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique.
For example, $3 = r^{-10} + r^{-5} + r^{-1} + r^2$ and $10 = r^{-10} + r^{-7} + r^6$.
Interestingly, the relation holds for the complex roots of the equation.
Let $w(n)$ be the number of terms in this unique representation of $n$. Thus $w(3) = 4$ and $w(10) = 3$.
More formally, for all positive integers $n$, we have:
$n = \displaystyle \sum_{k=-\infty}^\infty b_k r^k$
under the conditions that:
$b_k$ is $0$ or $1$ for all $k$;
$b_k + b_{k + 1} + b_{k + 2} \le 1$ for all $k$;
$w(n) = \displaystyle \sum_{k=-\infty}^\infty b_k$ is finite.
Let $S(m) = \displaystyle \sum_{j=1}^m w(j^2)$.
You are given $S(10) = 61$ and $S(1000) = 19403$.
Find $S(5\,000\,000)$.

This problem, as it has been stated, seems like it cannot be solved directly as you didn’t provide a pattern or formula to solve for $w(n)$. That means, we cannot directly find $S(5\,000\,000)$ unless we know the value of $w(n)$ for each $n$ in the range from $1$ to $5\,000\,000$.

However, let us try to break down the problem.

First of all, we need to understand that any number can be represented using the basis $r$ under the conditions you mentioned. Where $r$ is a real root of the equation $x^3 = x^2 + 1$.

Then follows a crucial insight that for a given positive integer $n$, we need to find the number of terms in its unique representation in base $r$, which you defined as $w(n)$.

Further, we know $S(m) = \displaystyle \sum_{j=1}^m w(j^2)$, that is, $S(m)$ is the sum of $w(j^2)$ for $j$ from $1$ to $m$.

To find $S(5\,000\,000)$, a direct approach would mean we need to solve for $w(1^2)$, $w(2^2)$, …, $w((5\,000\,000)^2)$, and sum up these values. In simpler words, we need the sum of the number of terms in the unique representation of the squares of numbers from $1$ to $5\,000\,000$.

Given $w(1^2)$, $w(2^2)$, …, $w(10^2)$ sums up to $61$ and $w(1^2)$, $w(2^2)$, …, $w(1000^2)$ sums up to $19403$, it is hard to find a general pattern or formula.

Without more information or taking a more approximate approach using statistics or perhaps machine learning, it seems impossible to solve this problem directly in a reasonable amount of time.

It’s however important to note that this problem might be solvable using certain combinatorial methods and equations relating to the series and sequence given. Another approach could be to find a function which approximates $w(n)$ given the information we have and use this to find an approximate answer.

More Answers: