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Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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9 votes
1 answer
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Is there a non-trivial arithmetic progression of positive integers such that every number contains the digit $2?$

Let $X:=\{$ positive integers that contain the digit $2\}$ For fixed $m,n\in\mathbb{N},$ define the A.P. $S_{m,n:=}\ \{m,\ m+n,\ m+2n, \ldots\}\ .$ I am interested in $S_{m,n}\cap X,$ and $S_{m,n}\cap ...
Adam Rubinson's user avatar
12 votes
2 answers
871 views

Numbers that are four times their decimal reverse

Given an integer $n \geq 4$, find all $n$-digit numbers which are 4 times the reverse of their decimal digits e.g. $8712 = 4 \times 2178$. For each value of $n$, there is only one solution. An ...
1123581321's user avatar
  • 1,102
1 vote
0 answers
30 views

Existence of $n$ where $S_b(n^k) \equiv r \pmod{M}$ where $S_b$ denotes sum of digits in base $b$

Let $b, k, M \in \mathbb{N} \setminus \{1\}$, $r \in \{0, 1, \dots M-1\}$ and $S_b: \mathbb{N}_0 \rightarrow \mathbb{N}_0 $ denote the method which outputs the sum of digits of its input in base $b$. ...
EnEm's user avatar
  • 1,181
-2 votes
1 answer
86 views

How can 0.2999... be exactly 0.3 [duplicate]

Maslanka asked a bizarre Pyrgic question on the 4th of May (see below (I had to type it in because I'm not allowed to post images yet)). What I couldn’t get my head around was that, as I understand it,...
Mike Gibbs's user avatar
2 votes
1 answer
283 views

This weird algebra number theory question

Find all positive integers $n$ such that: $n = \prod_{k=0}^{m} (a_k +1)$ where $a_ma_{m-1}...a_1a_0$ is decimal representation of $n$. Now I tried for $m=0$ (Trivially Not possible), $m=1$ : $a_1a_0 =...
CLASH ROYAL's user avatar
2 votes
1 answer
26 views

If $u$, $v$ and $w$ are the digits of decimal system, then the rational number represented by $0.uw\overline{uv}$ is?

Context I was recently giving a math test and encountered the following question: If $u$, $v$ and $w$ are the digits of decimal system, then the rational number represented by $0.uw\overline{uv}$ is? ...
Sambhav Khandelwal's user avatar
3 votes
0 answers
122 views

Forced factors of numbers like $100001000110002\cdots 10447$?

In an old factoring project I currently attack again I dealt with positive integers that emerge if one writes down the numbers $10\ 000$ to some number $k$ with $10\ 000\le k\le 99\ 999$ in increasing ...
Peter's user avatar
  • 85.1k
3 votes
2 answers
183 views

Perfect cubes with digit-average at least $7.5$

I found so far the following perfect cubes with a digit-average (in base $10$) with at least $7.5$ : ...
Peter's user avatar
  • 85.1k
8 votes
4 answers
3k views

Is 1/3 included in the sequence 0.3, 0.33, 0.333,...? [closed]

I assume that $\frac{1}{3}$ is equal to $0.3333...$. Let's define a sequence as follows: $0.3$, $0.33$, $0.333$, $0.3333$,... Question: is $\frac{1}{3}$ included in this sequence? Every item in the ...
psmith's user avatar
  • 247
12 votes
1 answer
168 views

$n^k$ with no $1$'s for $k \le 8$.

$n = 2975$ is the least positive integer with the property that none of $n, n^2, n^3, \ldots, n^7$ contain the decimal digit $1$: $$ \eqalign{2975^2 &= 8850625\cr 2975^3 &= 26330609375\cr 2975^...
Robert Israel's user avatar
0 votes
1 answer
73 views

Sum of digits (upto single digit) of a prime number raised to an even power

I noticed that sum of digits (upto a single digit) of a prime number (except 3) raised to an even power is always either 1, 4 or 7 (3 is an exception. 3 raised to any power always gives the sum of ...
Sundararajan V's user avatar
0 votes
0 answers
62 views

Probability of All Decimal Digits Appearing in a Ten-Digit-Block of Pi Assuming Normality

In their book A Biography of the World's Most Mysterious Number, Posamentier and Lehmann mentioned that "Conway has indicated that if you separate the decimal value of $\pi$ into groups of ten ...
mathy_mathema's user avatar
-1 votes
2 answers
116 views

Find the perfect squares of four digits whose square root is the sum of the numbers obtained if we separate the first two digits from the last two

the question We are asking for the perfect squares of four digits whose square root is the sum of the numbers obtained if we separate the first two digits from the last two the idea let the number be $...
IONELA BUCIU's user avatar
12 votes
4 answers
2k views

Can a decimal that is infinitely repeating in one base be nonrepeating in another?

For instance, can a number like $0.1111111\cdots$ in base $3$ be represented as $0.23515613\cdots$ (non-repeating) in base $8$? I imagine the answer would be a resounding NO but it would be ...
theboombody's user avatar
0 votes
0 answers
10 views

How to correctly report significant digits in measurements and their uncertainty?

I am calculating the stream density S (= river length L divided by stream area A). The river has a length L of 671.8 +- 5.9 km and the stream area is 1860.2 +- 19.8 km^2. The stream density is thus ...
Yoni Verhaegen's user avatar
0 votes
1 answer
97 views

The product $abc$ when $a+b+c=191$ [closed]

Find the greatest possible number of back-to-back zeros at the end of the product of three counting numbers if the sum of these three numbers equals 191. (The number 202100 has exactly 2 back-to-back ...
Maulish Soni's user avatar
6 votes
3 answers
274 views

How to prove that for all digits there exists a transcendental number which contains it infinite number of times in its representation?

Soon I was told a statement that for all digits a transcendental number can be found, containing the digit infinitely many times. It seems obvious, but cannot find good enough argument for it. Can ...
Az Sym's user avatar
  • 63
2 votes
2 answers
128 views

Does $\sum\limits_{n=1}^\infty\frac{1}{\text{Sum of permutations of digits of }n}$ converge?

Hopefully the following chart explains some things: $$\begin{array}{|l|l|} \hline n & \frac{1}{\text{Sum of permutations of digits of }n} \\ \hline 1 & \frac{1}{1} \\ \hline 2 & \frac{1}{2}...
Dylan Levine's user avatar
  • 1,688
0 votes
0 answers
26 views

What is the algorithm to represent any number in place-value subtrahends?

I'm going to ask a rather basic, but still curious question about arithmetic. It's a widely known fact that any number has a unique representation in place-value summands. For example, a number $137$ ...
Rusurano's user avatar
  • 846
1 vote
1 answer
47 views

Arithmetics with decimal numbers

Let's suppose that $$10,a+15,ba +15,3 = 2\times (20, ab)$$ where numbers are in their decimal representation, and so $ab$ and $ba$ are two digit numbers. Is there a straightforward way to evaluate $a\...
user avatar
2 votes
1 answer
144 views

Looking for an algebraic number with a balanced sequence of digits

It is generally believed that all irrational algebraic numbers $\alpha$ are normal, in all bases. In base $2$ that implies that there are arbitrary large $n$ such that for the binary expansion $$\...
orangeskid's user avatar
  • 54.9k
1 vote
0 answers
38 views

Expected value of index of an n-digit number found in pi

Let $E(n)$ be the expected value of the index of finding an n-digit number in the digits of pi e.g. Number 0 is found at index 32 Number 1 is found at index 1 Number 2 is found at index 6 Number 3 is ...
Winter's user avatar
  • 936
1 vote
1 answer
121 views

Is it possible to calculate the first digits of this number?

Is it possible to calculate the leading decimal digits of $3 \uparrow\uparrow 5\ = 3^{3^{3^{3^3}}} = 3^{3^{7,625,597,484,987}}$? Using currently known methods, this would require knowing the complete ...
Allam A.'s user avatar
  • 229
2 votes
1 answer
55 views

How is this summation expression transformed?

I am solving one math problem. I could not understand the following transformation described below. I am guessing the denominator is re-written as a factorial of 2K and unnecessary even terms are ...
Sherlock_Hound's user avatar
0 votes
1 answer
41 views

Solution to $x^2 + y^2 = x' + y'$ where $x'$ and $y'$ have the same digit-representation as $x$ and $y$ but with a trailing digit $2$ respectively.

So I was thinking of a particular problem the other day and started experimenting with it. It goes like this: I want to find natural numbers $x$ and $y$ such that: $x$ is an $n$-digit number, and $y$ ...
Victor Galeano's user avatar
0 votes
1 answer
56 views

Integers not containing a fixed substring in their decimal representation

Prove that for any fixed string of digits $S$ (each digit from $0$ to $9$), there are at most $o(N)$ integers in $\{1,2,\ldots,N\}$ such that their decimal representation does not contain $S$. This is ...
DesmondMiles's user avatar
  • 2,733
1 vote
2 answers
148 views

First digits of the iterated powers of 2

I wanted to show that the first digits of $(2^{2^j})_{j=1}^\infty$ are not periodic. By the standard Dirichlet trick I can show that any array of digits forms the initial digits of some number of the ...
Robert Barg's user avatar
1 vote
2 answers
103 views

Prove that any positive integer has a multiple whose decimal expansion involves all ten digits. [closed]

This is something I've been struggling with - I've been having trouble even wrapping my head around what is being asked. It feels like the answer might be on or around something extremely obvious, but ...
Jiles Mocon's user avatar
0 votes
0 answers
60 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....
Alexander Kalian's user avatar
3 votes
0 answers
106 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 ...
Allam A.'s user avatar
  • 229
0 votes
0 answers
47 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 ...
Lục Trường Phát's user avatar
4 votes
1 answer
158 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 $28$...
dckjavis's user avatar
3 votes
2 answers
314 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 $a_1$ ...
IONELA BUCIU's user avatar
2 votes
1 answer
86 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 ...
Matteo Bernasconi's user avatar
0 votes
1 answer
240 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 ...
absolut_jay's user avatar
2 votes
1 answer
98 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 ...
Peter's user avatar
  • 85.1k
0 votes
0 answers
29 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 ...
Michael T Chase's user avatar
0 votes
0 answers
25 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 &...
Sergey6552's user avatar
2 votes
1 answer
129 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 : $37= 1^3\times10+3^3$ ...
Pruthviraj's user avatar
  • 2,697
1 vote
1 answer
39 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 ...
Wagner Martins's user avatar
5 votes
3 answers
352 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=\...
shmurda's user avatar
  • 51
2 votes
1 answer
74 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 ...
ImHackingXD's user avatar
  • 1,090
1 vote
0 answers
40 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 ...
Connor James's user avatar
1 vote
0 answers
60 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 ...
Peter's user avatar
  • 85.1k
8 votes
4 answers
1k 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, ...
Riemann's user avatar
  • 717
5 votes
2 answers
446 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 ...
Etienne's user avatar
  • 87
0 votes
0 answers
22 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 $n,k$ , $k$ ...
Peter's user avatar
  • 85.1k
3 votes
2 answers
162 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 ...
Christopher Miller's user avatar
0 votes
0 answers
31 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 ...
ʈɦɘ ʙɑɕʞ's user avatar
5 votes
1 answer
349 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 ...
Max Muller's user avatar
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