24 questions linked to/from Starting digits of $2^n$.
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### Show that there are infinitely many powers of two starting with the digit 7 [duplicate]

This is a contest math problem which I was not able to solve. A hint toward the solution would be helpful as well. Problem: Show that there are infinitely many powers of 2 starting with the digit 7. ...
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### Is $2^k = 2013…$ for some $k$? [duplicate]

I'm wondering if some power of $2$ can be written in base $10$ as $2013$ followed by other digits. Formally, does there exist $k,q,r \in \mathbb N$ such that 2^k=2013 \cdot 10^q+r \,\,\,; \,\,\,r&...
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### Is it known or new? [duplicate]

Possible Duplicate: Starting digits of 2^n While I was randomly working with number patterns, I came along with some interesting pattern which seems to turn to a conjecture in fact. My ...
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### Does $\exists n \in \mathbb{Z}$ such that $2^n$ can start with $9786543120$? [duplicate]

I have tried $2^n=\displaystyle\sum_{k=0}^{n}\binom{n}{k}$ but could not reach further. Thank you.
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### Decimal expansion of $2^n$ [duplicate]

i am trying to read / understand the book: Introduction to the modern theory of dynamic systems, hassleblatt boris and katok anatole. On page 28 there is an excercise which I am trying to solve but ...
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### problem about number and power of 2 [duplicate]

Possible Duplicate: Starting digits of 2^n can anyone give me a hint ? Prove that any finite sequence of digits is a starting sequence of digits for some power of $2$. This is my attempt : ...
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### Proving the existence of an integer $n$ such that $2^n=123456789…$ [duplicate]

How to prove that it exists an integer $n$ such that the decimal expansion of $2^n$ starts with $123456789$?
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### Consider the sequence $1,2,4,8,…,a^n = 2^n,…$ of all the powers of $2$ [duplicate]

Consider the sequence $1,2,4,8,...,a^n = 2^n,...$ of all the powers of $2$. Prove that, given any digit $i ∈ {1,...,9}$, there exist infinitely many values of $n$ for which $a^n$ starts with that ...
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### Given $n$, exist $k$ such that $2^k$ contains $n$ as string. [duplicate]

I have this doubt: Given $n\in \mathbb{N}$, does exist $k\in \mathbb{N}$ such that $2^k$ contains $n$ as a string in it? For example, $53$ is in $2^{16}=65\color{red}{53}6$. I just thought the ...
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### Do the Fibonacci numbers contain any run of digits?

Related to this question and inspired by this challenge on PPCG. The challenge is as follows: for a given natural number $x \in \mathbb{N}$, find the first Fibonacci number $F_n$ that contains $x$. ...
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### Fractional part of $b \log a$

From the problem... Find the minimal positive integer $b$ such that the first digits of $2^b$ are 2011 ...I have been able to reduce the problem to the following instead: Find minimal $b$ such ...
### Is it possible that $3^n$ starts with $2019$ for some positive integer? [duplicate]
Does there exist a positive integer $n$ such that the decimal representation of $3^n$ starts with $2019$? My attempt: I have tried some starting powers and I guess no such positive integers exists. ...