Why $82000$? Numbers that can be written from base $2$ to base $5$ using only the digits $0$ and $1$

Question

This is really very curious. Many links on http://oeis.org/A146025 about this but -- why? I mean, this is not some abstract mathematical notation but rather something inherent in, I dunno, the structure of the world? Addition, multiplication and powers of positive integers are very "lightweight" abstractions, most of us probably learned it in elementary school and yet -- $82000$. Why? What's the deep structure hidden behind this mind boggling result? Like the Riemann zeta hiding behind the Basel problem, sort of.

Edit: to make this more clear. The Basel problem turned out to be the value of the Riemann zeta at 2. Is there a function which evaluates at 2,3,4,5 to 2,3,4,82000 and has some more deep meaning? Alternatively is there some other deeper problem for which the pairs 2-2, 3-3, 4-4, 5-82000 is a solution? Is there an explanation of sorts why this huge jump for 5? Dimensions perhaps?

Answer 1

I have my own answer. Just today I was reading What Just Happened: A Chronicle from the Information Frontier by James Gleick. When it talks about the Kolmogorov–Chaitin complexity it mentions:

Some mathematical facts are true for no reason. They are accidental, lacking a cause or deeper meaning.

Whether there's more to 82000 than "the smallest number that can be written with just 0 and 1 digits in bases 2, 3, 4, 5" (and perhaps the only one in 2, 3, 4, 5 and also perhaps the only one for any 2, 3, ....N where N > 4) simply can not be answered. Maybe in the future there will be another meaning but maybe not. This is closely related to the halting problem and of course Gödel's incompleteness theorems. For the sake of archives, the easiest introduction into the latter is in the Monte Carlo Lock chapters of Raymond Smullyan's The Lady or the Tiger. I understood that as a high schooler. And sorry for not realizing my question touched this, my exams in these at the university have been twenty years ago and haven't dealt with these topics much since.

Answer 2

Let's approximately model the digits of a number in positional notation with coprime bases as independent random variables, and likewise for the number's residues with respect to powers of different primes, with mutual conditional independence. (I think you could make this more rigorous by uniformly randomly selecting a number between $1$ and $N$ and taking the limit $N\to\infty$.)

Let $E_{p^k}$ be the event that the number is divisible by $p^k$, and let $B_{bj}$ be the event that the number's $j$-th digit in base $b$ is $0$ or $1$.

Then

\begin{align} \def\Pr#1{\textsf{Pr}\left(#1\right)}\Pr{E_{2^3}\cap E_{5^3}\mid\bigcap_{b=4}^5\bigcap_{j=1}^3B_{bj}} &= \Pr{E_{2^3}\mid\bigcap_{j=1}^3B_{4j}}\Pr{E_{5^3}\mid\bigcap_{j=1}^3B_{5j}} \\ &=\frac14\cdot\frac18 \\ &=\frac1{32}\;. \end{align}

So in some sense the "probability" that if there is only a single non-trivial number with this property it would end in three zeros "was" $\frac1{32}$.