Largest power of 2 which doesn't contain 0?

What is the largest power of $2$ which doesn't contain $0$ ?

I tried a variety of methods to tackle this problem including programming and I even got the answer as $86$ but I couldn't prove that there exists no larger power of $2$ which doesn't contain $0$ .

Note: I tried brute forcing for integers up to ${2}^{100000}$ but with no luck. So for ${2}^{n}$ , the $n$ must be greater than $100000$ . Inspiration (Calvin Lin's comment).

#Algebra #Combinatorics #NumberTheory #Exponents #ComputerScience

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

If you've tried up to $2^{100000},$ which has in excess of $30000$ digits, then the probability that any given power of two greater than this not including a $0$ would be less than $10^{-1378}.$ While this is not a proof, it seems pretty certain that no greater power than $2^{100000}$ will be devoid of a $0.$ So perhaps $86$ is indeed the solution that you are looking for.

Edit: This is in fact the conjectured greatest power with this property, but a proof remains an open problem.

Brian Charlesworth - 5 years, 11 months ago

Relevant.

Pi Han Goh - 5 years, 7 months ago

99999999999999999999998

Rushikesh Jyoti - 5 years, 10 months ago

Prove it.

Arulx Z - 5 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Largest power of 2 which doesn't contain 0?

Note: I tried brute forcing for integers up to 2100000{2}^{100000}2100000 but with no luck. So for 2n{2}^{n}2n, the nnn must be greater than 100000100000100000. Inspiration (Calvin Lin's comment).

Comments

Note: I tried brute forcing for integers up to ${2}^{100000}$ but with no luck. So for ${2}^{n}$ , the $n$ must be greater than $100000$ . Inspiration (Calvin Lin's comment).