Questions tagged [decimal-expansion]
For questions about decimal expansion, both practical and theoretical.
1,225
questions
0
votes
0
answers
39
views
Can the occurrence of certain digits be proven/disproven, for any arbitrary irrational number?
$\pi$ is perhaps the most famous irrational number. We know it contains all decimal digits from 0-9, just by virtue that all digits occur, at least once, within 32 decimal places:
$\pi = 3....
2
votes
0
answers
42
views
Is there a better way to compute first digits of very large numbers?
The currently known method of finding the first digits of $a^b$ is multiplying $\log_{10} a$ by b, and extracting the fractional part. This allows us to compute the first digits of quite large numbers ...
0
votes
0
answers
46
views
Prove that: $n$ has a prime divisor which is not smaller than $11$.
Let integer $n>10$ such that $n=\overline{a_{m}a_{m-1}a_{m-2}...a_{0}}$ where: $a_i \in {(1;3;7;9)}$, $i=\overline{0,m}$
Prove that: $n$ has a prime divisor which is not smaller than $11$.
Here or ...
4
votes
1
answer
128
views
Does anyone know why this is happening with 1/7? [duplicate]
(I've never posted on here before, so apologies for any formatting problems)
I had always noticed this property of the decimal form of $1/7$ ( $0.14285714...$ ) where the decimals went $14$ , then $28$...
2
votes
2
answers
277
views
Show that the number $A$ is irrational
question
For each $n$ natural number we denote by $a_n$ the first digit a
of the number $n^3$.
Show that the number $A = 0, a_1a_2 ... a_n ...$ is irrational.
my idea
A thing is clear....between $a_1$ ...
2
votes
1
answer
81
views
Show decimal expansions
I can't wrap my head around this exercise:
Show that the rational number $\frac 94$ has two different decimal expansions, namely $2.2500000\dots$ and $2.2499999\dots$ by writing these decimal ...
0
votes
1
answer
206
views
I’ve observed an interesting pattern where the last digit of the repeating decimal sequence of 1 / prime
I’ve observed an interesting pattern where the last digit of the repeating decimal sequence of 1 / prime
1/prime matches the last digit of the prime number itself for several primes. This pattern ...
2
votes
1
answer
75
views
Is $\omega(n)=16$ the maximum?
What is the largest possible value of $\omega(n)$ (the number of distinct prime factors of $n$) , if $n$ is a $30$-digit number containing only zeros and ones in the decimal expansion.
I checked ...
0
votes
0
answers
18
views
Decimal exponent
This is how I've attacked 3^1.0987, doing it with pen and paper. 3(3)^(987/10000) = 3*3^(((3)(7)(47))/10000). But from here I don't know what to do. I can't imagine trying to take the 10,000th root ...
0
votes
0
answers
23
views
Sum of squared digits
Let $s(n)$ denote the sum of the squares of the digits of $n$. For example, $s(14) = 1 ^ 2 + 4 ^ 2 = 17$ Determine all integers adding n for which $s(n) = n$ holds.
I bound it to $243$ due to $9^2 *4 &...
1
vote
1
answer
121
views
Representing number, where digits are cubes
Let $n$ is integer and we want to express it in terms of $10$ as $$n=R^3_k10^k+R^3_{k-1}10^{k-1} \cdots+ R^3_0$$ where $R_i\in \{\pm0,\pm 1,\pm 2,\ldots,\pm9\}$
Example : $37= 1^3\times10+3^3$
...
1
vote
1
answer
30
views
Approximate summation formula of time to count numbers from 1 to N
When calculating how much time it takes to count from $1$ to $n$, it is normally used the approximation that it takes about $1s$ to say a number out loud, so it would take $n$ seconds, but there's a ...
5
votes
3
answers
344
views
Arbitrary decimal value of $A(n)=\left(\frac{11}{10}\right)^n$
For $n\in\mathbb{Z}$ consider the number
$$A(n)=\left(1+x\right)^n\bigg{|}_{x=\frac{1}{10}}=\sum_{k=0}^\infty\binom{n}{k}10^{-k}$$
which we have expanded by the Taylor series. It is found that
$$a_1=\...
2
votes
1
answer
61
views
Probability that the first $m$ digits in $2^n$ are $k_1k_2\dots k_m$
I want to find the probability that the first $m$ digits of powers of 2 are a given combination $k_1k_2\dots k_m$. So far, here's my reasoning:
A number $2^n$ will have the first $m$ digits of the ...
1
vote
0
answers
36
views
Deleting Digits from Champernowne's Constant
As some may know, Champernowne's constant is one of the only known constants proven to be normal. The number is constructed by concatenating whole numbers as you count up and appending them behind a ...
1
vote
0
answers
45
views
Two questions about emirp's
Define $r(n)$ to be the reverse of a positive integer , that is the number emerging if the decimal expansion is written down in reverse order. Emerging leading zeros are of course omitted , but this ...
4
votes
0
answers
262
views
Is there a known explanation for the Feynman point?
The Feynman point is a mathematical coincidence. It states that from position 762, there are six consecutive nines in the decimal expansion of pi. Some mathematical coincidences have an explanation, ...
5
votes
2
answers
421
views
Similarities in the digits of the powers of 2 and 5
Many may have noticed that the negative powers of 5 contain the same digits as the positive powers of 2:
This pattern intrigued me. I started to wonder if it exists in different number bases. I soon ...
0
votes
0
answers
19
views
Is this the smallest Proth - emirp of the desired form?
An emirp is a prime number that keeps prime if the digits in base $10$ are written down in reverse order. A Proth-prime is a prime number of the form $2^n\cdot k+1$ with positive integers $n,k$ , $k$ ...
3
votes
2
answers
137
views
Estimating how many of the first $10,000$ Fibonacci numbers start with the digit $9$
Consider the problem of estimating how many of the first $10,000$ Fibonacci numbers begin with the digit $9$.
The only ideas I have so far:
Obviously, if we assume that the every first digit is ...
0
votes
0
answers
28
views
What is the chance of getting same decimals of percentage 4 times in a row with RNG?
I was rolling some random percentage with RNG between 0 and 1; if you multiply the result to 100, you're getting the percentage between 0 and 100, decimal amount was 4. And then I got 0.7171% the ...
5
votes
1
answer
342
views
Do permutations on the decimal expansions of irrational numbers retain the property of irrationality?
Suppose we have an irrational number with the following decimal expansion:
$$A = a_0 \ a_1 \ a_2 \ a_3 \ a_4 \ a_5 \ a_6 \dots $$
Now, construct a new real number through a permutation on the decimals ...
5
votes
0
answers
87
views
Irrational Numbers and Surjection from $[0,1)$ to $[0,1)^2$
I am searching for some clues or solutions of the question below:
For $\sqrt{2}=1.41421356\cdots$,
Is $1.1236\cdots$ irrational?
To say more formally:
Let $f$ a function from $[0,1)$ to $[0,1)^2$ ...
1
vote
0
answers
31
views
two forms of infinite decimal which are rewritten from finite decimal
How to transform a finite decimal into a infinite decimal? It sounds nothing to discuss. But I have seen two forms to deal with it which make me confused.
Let's consider a finite decimal $x_0.x_1x_2\...
0
votes
0
answers
42
views
If $\{n_k\}$ is the set of natural numbers with no 0 in their decimal expansion, $\sum_{k=1}^\infty \frac{1}{n_k}$ converges to a number less than 90 [duplicate]
Let ${\{n_1,n_2,…\} }$
be the set of natural numbers that do not use the digit 0
in their decimal expansion. Then, the series
$\sum_{k=1}^\infty \frac{1}{n_k}$
converges to a number less than 90.
Is ...
3
votes
1
answer
91
views
What is the measure of the set of numbers in $[0,1]$ whose digits have a mean of $0$?
I've been playing around with ways to systematically define a continuum of dense and uncountable subsets of real numbers in a (somewhat) intuitive manner, and tried the following characterization:
For ...
2
votes
1
answer
170
views
How many subsets of $\{0,1,2,3,4,5,6,7,8,9\}$ could be realized as the set of distinct digits of a prime?
How many subsets of $\{0,1,2,3,4,5,6,7,8,9\}$ could be realized as the set of distinct digits of a prime?
Possible solution
Obviously, the empty subset is not realizable.
Five singleton subsets are ...
1
vote
0
answers
35
views
Carmichael-numbers with only one odd digit
Here I ask for a third Carmichael number with only odd digits in their decimal expansion. Far more Carmichael numbers seem to exist with the property that in the decimal expansion there is only one ...
4
votes
1
answer
162
views
Numbers such that $(\overline{a_1\dots a_n})^2=\overline{x_1\dots x_m}$ and $(\overline{a_n\dots a_1})^2=\overline{x_m\dots x_1}$.
Recently, I had the pleasure of finding out that
$$13^2=169\quad\text{and}\quad 31^2=961.$$
It had me wondering . . .
The Question:
What pairs of distinct natural numbers $r,s$ have decimal ...
1
vote
1
answer
80
views
Prove that This Property Does Not Hold for Any Other Pair of Digits.
Given a positive integer $n,$ prove that there is a positive integer $m$
that to base ten contains only the digits $0$ and $1$ such that $n|m.$ Prove that the same holds for digits $0$ and $2,$ or $0$ ...
2
votes
0
answers
153
views
Is there a third Carmichael number with only odd digits?
Upto $2^{64}$ , there are two Carmichael numbers with only odd digits :
$$53711113=157\cdot 313\cdot 1093$$ and $$3559313513953=29\cdot 113\cdot 337\cdot 673\cdot 4789$$
In the first case, the prime ...
1
vote
1
answer
757
views
Is there any perfect power in the sequence $12,123,1234,12345,...$?
Inspired from the question Is there any perfect square in the sequence $12,123,1234,12345,...$?, there is no perfect square other than $1$ in the sequence of Smarandache numbers. But I wonder if are ...
1
vote
1
answer
23
views
To prove that the Cantor function maps the end points of the intervals removed to a same point
For the function f in 1.6D i.e. $$x=0.b_1b_2....(3)$$ then
$$f(x) = y= 0.a_1a_2.....(2)$$ where $a_i=b_i/2$
Here (2),(3) represents the binary and ternary expansions of $x$ respectively.
Show that if $...
0
votes
1
answer
267
views
What numbers have a multiple that in base 10, can be written with only the digits $2$ and $5$.
I was working on the classic number theory problem "for any integer $n$, it has a multiple whose base $10$ representation consists of only $1$s and $0$s" (for anyone who stumbles on this ...
1
vote
0
answers
33
views
Proportion of a digit in an algebraic number's binary expansion
We know that
A real number is rational if and only if it's binary (or base $n$ expansion, for all $n$) is eventually periodic. Therefore, the proportion of each digit (0 or 1 in the binary case) is a ...
0
votes
1
answer
89
views
Perfect square in the sequence formed by consecutive numbers
I recently saw a question Is there any perfect square in the sequence $12,123,1234,12345,...$?
This led to thinking about a new question.
Consider sequence https://oeis.org/A057137 that is the ...
1
vote
2
answers
85
views
Find the smallest positive multiple of $1999$ that ends in $2006$ (last four digits)
Find the smallest positive multiple of $1999$ that ends in $2006$ (last four digits)
Approach:
$1999N\equiv 2006\pmod{10000}$
(1) $9995N\equiv-5N\equiv30\pmod{10000}$
(2) $-N\equiv 6 \pmod{2000}$, so ...
0
votes
0
answers
26
views
Let $t$ be a positive real number. Prove that there is a positive integer $n$ such that the decimal expansion of $nt$ contains a $7$.
From this question, I'm honestly not too sure on how to prove it.
I've thought about an approach, which is to take 3 numbers from t. Then prove that the number in the middle of the product of those 3 ...
0
votes
0
answers
44
views
why does the sequence of n/7 always have the same six digits in the same order in the decimal part?
I realized the other day that in the decimal representation of $\frac{n}{7}$, where $n=\{1,2,3,4,5,6\}$, the decimal part always has the same sequence of six digits but shifted to start at a different ...
1
vote
0
answers
50
views
Analogue to emirp for Carmichael numbers
An emirp is a prime number with the property that if we write down the digits in base $10$ in reverse , we again get a prime number. Trivial emirp's are the palindrome primes (if we write down the ...
1
vote
2
answers
204
views
Proof that a repeating decimal has non-repeating digits after decimal iff denuminator has factors of 2 or 5 besides other prime factors
I'm studying a lesson about fractions. It classifies rational numbers into three categories based on their decimal representation.
Terminating Decimal: If a reduced fraction'denuminator has only ...
3
votes
1
answer
251
views
How many numbers less than N have a prime sum of digits?
I'm working on solving Project Euler's Problem 845. It's asking us to find the $10^{16}$-th positive integer number that has a prime sum of digits. Adopting a 'naive' solution, I compute the sum of ...
2
votes
1
answer
72
views
Formula for calculating the difference between sums of digits in the same base for $(n-1)$ and $n$
I am looking for a formula to calculate the difference between the sums of the digits of $(n-1)$ and $n$ in a given base, denoted as $sb(n-1) - sb(n)$. Specifically, I want to find a formula that ...
2
votes
1
answer
73
views
Calculating the length of a decimal expansion in constant time
Is there a way to calculate the length of a decimal expansion as a result of a division operation in constant time?
$\frac{1}{256} = 0.00390625$ therefore the expansion length is $8$.
$\frac{1}{357} = ...
3
votes
0
answers
115
views
Proving that there are infinite primes with digit sum 8 in base 10
I recently wrote about a problem I cam up with while thinking about number theory, which you can find on this post. Long story short, I'm trying to prove there are infinite natural numbers such that ...
1
vote
2
answers
157
views
Call $n\in\Bbb N$ "balanced" if the sum of its digits equals the count of its divisors. How many "balanced" numbers are there up to $m$?
I recently stumbled across a problem about numbers' divisor count (more specifically, how many positive integers are equal to the square of their divisor count - answer was 2: they are 1 and 9).
But I ...
4
votes
2
answers
195
views
Proof of $\inf\left \{ \frac{\mathrm{d} (n^2)}{\mathrm{d} (n)} \; \bigg| \; n \in \mathbb{N} \right \}=0$, where $d(n)$ is the sum of digits of $n$
So I wanted to find the infimum of the set described in the title, and I'm pretty confident on what the subsequence should be to ensure a 0 infimum.
But a proof of this eludes me. I tried some funky ...
2
votes
0
answers
170
views
Question about the collection of the prime factors of a fibonacci number
A positive integer $n$ is called pandigital , if every digit from $0$ to $9$ occurs in the decimal expansion of $n$.
Conjecture : The largest non-pandigital fibonacci-number (a fibonacci-number with ...
0
votes
1
answer
78
views
Useful length of Pi?
Not sure where this really fits, so am trying Mathematics first. Feel free to migrate to another StackExchange forum if more appropriate elsewhere. So I was listening to a podcast yesterday that was ...
1
vote
0
answers
31
views
Is there a name for such rounding algorithm?
When I compute a number where an approximation is needed but no accuracy is specified, I usually scan the first few digits and if I find a zero, then I truncate the decimals before the first zero. For ...