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I wanted to know if there is a mathematical notation to say verbally "it does not include the digit zero in the number".

For example, $86$ is (conjectured to be) the largest integer $n$ such that $2^n$ does not include zeros. (FYI: $2^{86} = 77371252455336267181195264$.) Is there a better way to express this than $$86=\max\{n \mid 2^n \;\text{doesn't include "zeros" in the result}\}$$

Thank you very much!

Blue
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Francesco
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    mathematical notation rarely deals with such down-to-earth things as the representation in your favourite base that happens to be ten – Hagen von Eitzen May 19 '20 at 17:36
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    Btw, is the underlying claim true? Of course, we can test this up to high $n$ until it becomes extremely unlikely that such a ling number does not have a digit zero in it ... – Hagen von Eitzen May 19 '20 at 17:37
  • Welcome to math.se! See if your question is answered here, here, or possibly there. – Couchy May 19 '20 at 17:47
  • @HagenvonEitzen https://oeis.org/A007377 Says it's a long-standing open problem. – saulspatz May 19 '20 at 17:49
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    It's easy to fall into the trap of thinking that you need to express everything in notation. But it's easier to just say "it does not include the digit zero in its base-ten representation". Or you can invent your own word for this. Clarity is the important thing. – Jair Taylor May 19 '20 at 18:30

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Call the unique base-ten expansion of a finite number $X$ that doesn’t involve insignificant zeroes the series $$X = \sum_{k=a}^{b} 10^k x_k$$ Therefore, in base ten, the $k$th digit of $X$ is $x_k$ (depending on how you enumerate the digits, but this is not important).

Define $\newcommand{\Exp}{\operatorname{Exp}} \Exp$ to be the operator that returns the sequence $$\Exp X = (x_k)_{k=a}^{b}$$ Sequences are also denoted with braces as in $\Exp X = \{x_k\}_{k=a}^b$; the choice is yours.

Then in set notation we say $$0\notin\Exp X$$

For example, $$\Exp 2^{86} = (4,6,2,5,9,1,1,8,1,7,6,2,6,3,3,5,5,4,2,5,2,1,7,3,7,7)$$ The digits are reversed because in any series $a<b$. If you wanted to make the digits appear in the normal order, you would need to flip the limits of series and change $10^k$ to $10^{-k}$.

This tells us that there exists a sequence $(x_k)_{k=a}^b$ that does not contain $0$ that can be used in the series expansion of $X$. We can notate that statement as

$$\exists (x_k)_{k=a}^b \not\ni 0 : X = \sum_{k=a}^b 10^kx_k$$

and I bet that is what you’re really looking for.

gen-ℤ ready to perish
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Maybe not shorter, but potentially a more workable mathematical notation.

Let $2^n=a_0*10^0+a_1*10^1+...=\sum_i a_i10^i$ and let $A$ denote the set of {$a_i$}, then you conjecture that $86=\max\{n|a_i\neq0\forall a_i\in A\}$. This generalizes nicely to expressing the same statement in other base units.

Tyberius
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