Questions tagged [digits]
The digits tag has no usage guidance.
41
questions
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Can $N^2$ have only digits 0 and 1, other than $N=10^k$?
Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's?
It seems very unlikely,...
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34
votes
2
answers
2k
views
Is the sum of digits of $3^{1000}$ divisible by $7$?
Is the sum of digits of $3^{1000}$ a multiple of $7$?
The sum of the digits of $3^{1000}$ can be computed using a computer. It is equal to $2142$, so the answer is positive.
Is there a short proof ...
- 443
24
votes
0
answers
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views
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 ...
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16
votes
0
answers
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 \...
- 28.7k
15
votes
2
answers
1k
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The parity of the maximal number of consecutive 1s in the binary expansion of an integer
For an integer $n$, let $\ell(n)$ denote the maximal number of consecutive $1$s in the binary expansion of $n$. For instance,
$$ \ell(71_{10}) = \ell(1000111_2) = 3. $$
Consider the set $E$ of all ...
- 1,510
13
votes
1
answer
552
views
Can we compute the first $n$ digits of $\pi$ in $F(n)$ time?
I've seen various fast algorithms for computing the first few, or directly the $n$-th, digits of $\pi$.
However, it seems to me that all these algorithms assume (see last sentence here) that there are ...
- 18.7k
12
votes
0
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563
views
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 $\sum a_i p^i$...
- 28.7k
9
votes
1
answer
729
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Lower bound on # of nonzero digits in ternary expansions of powers of 2?
Does anyone know of any lower bounds on the number of nonzero digits that appear in powers of 2 when written to base 3? (Other than the easy "If it's more than 8 it has to have at least 3.") I know ...
- 2,495
8
votes
0
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413
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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 ...
- 181
6
votes
3
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685
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sum of binary and ternary digits
A problem in group theory (indices of imprimitive groups) gives rise to the following conjectures in number theory. Suppose a positive integer $n$ has binary and ternary expansions $n=\sum_{k\...
- 1,921
6
votes
1
answer
830
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How to explain this prime gap bias around last digits?
My question is related to this article by Oliver and Soundararajan (article about a bias in the distribution of the last digits of consecutive prime numbers).
After trying some python experimental ...
6
votes
2
answers
852
views
Does this sequence of ratios of digit sums have a limit?
I asked this question a few hours ago on MathStackExchange and there it received some attention but we still do not have a proof so I decided to ask it here also, in an unchanged form, and here it is:
...
user114642
6
votes
1
answer
690
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Optimal lower bounds for the sum of digits in base $b$
Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$
(e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...
user40023
6
votes
0
answers
202
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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{...
- 28.7k
5
votes
2
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Number of 1 in binary representation of n
Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.
For example, $13=1101_2$ so $1(13)=3\\$
Is there explicit form of $\,\,\sum{1(i)x^i} $?
I checked OEIS and didn't find ...
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5
votes
2
answers
981
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Is there a Bailey–Borwein–Plouffe (BBP) formula for e? [duplicate]
I recently used Bailey–Borwein–Plouffe formula to implement a π digit generator. Now I also want to implement an e digit generator, for the Euler number.
I've ...
user213
4
votes
1
answer
294
views
Can we always attain another prime via inserting digits between the digits of a fixed prime?
The sequence OEIS A080437 is
For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m.
I don't see why this ...
- 24.1k
4
votes
0
answers
121
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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 ...
- 19.1k
3
votes
2
answers
762
views
Density of the set of numbers whose sum of digits is prime
Let $A$ be the set of numbers whose sum of digits is prime (http://oeis.org/A028834).
I would like to know if $A$ has zero natural density, that is, if $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...
- 41
3
votes
1
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Normality property of powers of integers?
Inspired by this question, is there some conjecture stating that
$$
\limsup_{n \to \infty} \frac{d_j(2^n)}{dc(2^n)} = \frac{1}{10}
$$
where $d_j(m)$ counts the number of $j$s in the digits of $m$,
and ...
- 15.4k
3
votes
1
answer
124
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Golden ratio base
Let $\phi$ be the golden ratio and look at real numbers as expansions in digits from base $\phi + 1$. Has this base been considered or studied anywhere?
Note that integers in this base are palindromes ...
- 1,163
3
votes
0
answers
164
views
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 $0,0,1,2,4,5,7,9,12,13,15,\dots.$
...
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3
votes
0
answers
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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? ...
- 2,169
2
votes
1
answer
113
views
Measure of real numbers with converging average over binary digits
Consider the unit interval $[0,1]$, and by digits of $x\in[0,1]$ I mean its binary digits after the separator with no 1-period.
If $x_1,x_2,x_3,...$ are the digits of $x$, then consider the $k$-th ...
- 11.9k
2
votes
1
answer
702
views
Here is a generalization of n-ary base notation for numbers. Surely unoriginal. Anybody know where to find literature on it?
If $f:\mathbb{N}\to\mathbb{N}$ is any strictly increasing function with $f(0)=1$, define the base $f$ notation for natural numbers inductively as follows:
$0$ is represented as $()$ (the empty ...
- 121
2
votes
1
answer
87
views
Consecutive prime numbers in permutations of digits of the first consecutive positive integers
I have been toying for a while with the study of: in how many distinct primes and of which size can we divide permutations of digits of the first positive integers?
In this post I studied how many ...
- 289
2
votes
1
answer
87
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Partitioning integers into two parts and exploring relationships with positional numeral systems
I asked this question in Mathematics StackExchange (link) about a month ago, but I have received no answer. It is about the following problem:
Problem:
Are there sets $A,B$ of integers such that $A\...
2
votes
1
answer
224
views
The number of numbers no greater than n that are divisible by all their suffixes
My question: what a formula for finding the number of numbers no greater than n that are divisible by all their suffixes.
e.g: 5, 25, 125, 0125, 70125 are divisors of 70125.
refinement: $\overline{0....
2
votes
0
answers
75
views
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 , ...
user111524
1
vote
1
answer
54
views
Maximum product of digits of a perfect power
Today I found an interesting problem on the Internet - to find the exact power that has the largest possible product of digits.
Of course, we know that there are arbitrarily large squares that do not ...
1
vote
1
answer
203
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Runs of consecutive numbers that are not relatively prime to their digital sum
It is well known that there can be at most 20 consecutive integers (in base 10) that are divisible by their digital sum, so called Harshad or Niven numbers.
How long can a run of consecutive ...
- 3,497
1
vote
2
answers
321
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Square and reversed integer
For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$,
we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= \sum_{i=0}...
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1
vote
1
answer
350
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Problem related to inequality of sum of digits of power sum
Let $D$ be the function define as $D(b,n)$ be the sum of the base-$b$ digits of $n$.
Example: $D(2,7)=3$ means $7=(111)_2\implies D(2,7)=1+1+1=3$
Define $S(a,m)=1^m+2^m+3^m+...+a^m$ where $a,m\in\...
- 537
1
vote
0
answers
568
views
Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]
Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José).
My question is, can we ...
user141903
1
vote
0
answers
70
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On the sum of digits of primes in binary form [duplicate]
Let $s_2(m)$ be the sum of digits of $m$ in binary form.
I would like to ask the following question:
Is it true that for every $n\in \mathbb{N}$ there is at least one
prime $p$ which has $s_2(...
- 3,746
1
vote
0
answers
736
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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 $n$...
- 335
0
votes
1
answer
246
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Does a sequence of primes defined like this exists?
Does there exist a strictly increasing sequence of primes $(q_i)_{i \in \mathbb N}$ such that $\text {ds} (\prod_{k=1}^l q_k)$ is prime for every $l \in \mathbb N$?
Here $\text{ds}(n)$ denotes a ...
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0
votes
1
answer
126
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The series $\sum_{n=1}^\infty {2n\brace n}^{-{2n\brace n}}$ and $\sum_{n=1}^\infty (2n)_{n}^{-(2n)_{n}}$ in the context of normal numbers
In this ocassion we consider the followgin series that involve ${n\brace k}$ the Stirling number of the second kind and $(n)_k$ the Pochhammer symbols. I've known from an informative point of view ...
0
votes
0
answers
69
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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\...
- 3,058
0
votes
0
answers
81
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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, ...
-4
votes
1
answer
291
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Multiplicative Persistence - Highest persistence found? [closed]
tried to ask on the math reddit but got deleted due to my account being new.
Is the record for highest multiplicative persistence found still 11? As I may have just found a number with persistence of ...
- 21