Questions tagged [digits]
The digits tag has no usage guidance.
14
questions with no upvoted or accepted answers
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0's in 815915283247897734345611269596115894272000000000
Is 40 the largest number for which all the 0 digits in the decimal form of $n!$ come at the end?
Motivation: My son considered learning all digits of 40! for my birthday. I told him that the best way ...
18
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0
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2k
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Distribution of digits of $pq$-adic idempotents (aka "automorphic numbers")
Let $p$ and $q$ be distinct primes. By the ring of $pq$-adic integers I mean the ring $\mathbb{Z}_{pq} := \varprojlim \mathbb{Z}/(pq)^n\mathbb{Z}$ which is obviously isomorphic to $\mathbb{Z}_p \...
12
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Power series defined by Witt vectors / Teichmüller representatives of p-adics
Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i p^i$...
8
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Zero's in the decimal representation of powers of 3
This looked like an easy exercise, when a friend of mine asked me if I know a way to prove that the decimal representation of $3^k$ always contains a zero for $k\ge k_0$, but the more I think about ...
6
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208
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Choice of digits for extensions of $\mathbb{Q}_p$
I am interested in writing (in base $p$) elements of the maximal unramified extension $\mathbb{Q}_p^{\mathrm{unr}}$ of $\mathbb{Q}_p$, or (its completion) the field $\mathrm{W}(\mathbb{F}_p^{\mathrm{...
6
votes
1
answer
278
views
Sequence of digits of powers of two
Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational ...
4
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438
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There are infinitely many prime which have arbitrary large gap in their digits in particular base expansion
Consider $m$ and $r$ is any fixed positive integer and $t$ is a variable $(t=0,1,2,3,...)$. Below, $[a]$ denotes the greatest integer function of $a$ (or floor function).
Claim 1 : There exists a ...
3
votes
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167
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A recursion for the total number of 1's in binary expansions of the first natural numbers?
Let $$a(n)=a(2^k-n)+k(n-2^{k-1})$$ for $$1 \leqslant {2^{k - 1}} < n \leqslant {2^k}$$
with initial values $a(0)=0, a(1)=0, a(2)=1.$
The first values are $0,0,1,2,4,5,7,9,12,13,15,\dots.$
...
3
votes
0
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1k
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sum of digits in different bases
Given a natural number, What is the maximal natural number below it, whose sums of digits in base 10 and base 2 are the same? Is there a clever algorithm to do this aside from the brute force search? ...
2
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Powers of special class of positive integers whose representation in a base consists of digits only powers of that integer
For an integer $m \in (\sqrt {10} , 10)$ , define $A_{10,m}:=\{n \in \mathbb N : m^n=\sum_{j=0}^k 10^j m^{n_j} ; n_j=0 $ or $1; k \ge 0\}$ . So , $A_{10,m}$ is the set of those natural numbers , ...
1
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0
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738
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Kaprekar's mapping fixed points
Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine):
"Let $d(n)$ denote $n$...
0
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43
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How to find A(i, d)?
Let $s(n)$ denote the digit sum of a natural number $n$. For $i, d\in \mathbb{N}$ define $$A(i, d) = \limsup_{m\to \infty}\frac{|\{n\leq m | s(n)\equiv i\mod d\}|}{m}.$$ Compute $A(i, d)$ for all $i, ...
0
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70
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Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed
The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
0
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Generating the digits in a base system by repeated multiplication of a number
The first 15 terms of the sequence {a_i} = 2^i are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768. All of the digits in base-10, i.e. {0, ...