# Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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### How many zeros in the decimal representation of $5^n$?

I'm curious about some properties of the powers of 5 $$5^2=25,\quad5^3=125,\quad 5^4=625,\quad 5^5=3125,\quad ...$$ Is it true that at least $50$% of the digits in the decimal representation of $5^n$ ...
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### How are the Periods of the Decimal Expansions of $\frac{p}{q}$ and $\frac{q}{p}$ Related?

In an excellent post several years ago, we learn that the period of the decimal expansion of a rational number $\frac{p}{q}$ must divide the multiplicative order of $10\pmod q$ assuming that there are ...
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### What level of infinity is referred to when talking about recurring digits?

If a digit is written as $3.\dot{3}$, what level of infinity do the dots continue on for? Can this be proven to be true, or is it just a quirk of the notation? More specifically, $1/3$ can obviously ...
39 views

### What is known about representations of numbers of the form $\frac{1}{n}$, where $n$ is odd?

I calculated some decimal representations of numbers of the form $\dfrac{1}{n}$, where $n$ is odd, and most of them had a period that began immediately after the decimal point. One example of a number ...
3k views

### Extending prime numbers digit by digit while retaining primality

I looked at a table of primes and observed the following: If we choose $7$ can we concatenate one digit to the left so as to form a new prime number? Yes, concatenate $1$ to obtain $17$. Can we do ...
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### Asymptotic formula for “Self Numbers” $s_n$

While digging through some old notebooks today, I found a problem from a long time ago that I was never able to solve. It involves a sequence of positive integers called the “self numbers” defined ...
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### Is $1 + \lceil{\log_{10}x}\rceil = \lceil{1 + \log_{10}x}\rceil$ can anyone please prove or disprove it. [closed]

it seems to be true, but I dont have the proof
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### Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?

While discussing prime powers and divisors, I came up with the following problem. Examples $\to$ prime $p=3$ digit sum (in base ten) of $p^3=27$ is $p^2=9$, a power of $p$,. $\to$ prime $p=7$ ...
2k views

### Are there any irrational/transcendental numbers for which the distribution of decimal digits is not uniform?

I conjecture that for irrational numbers, there is generally no pattern in the appearance of digits when you write out the decimal expansion to an arbitrary number of terms. So, all digits must be ...
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### What reason is there to conjecture that every finite string is really in the decimal expansion of $\pi$?

One of my students asked me this, and it occurred to me that I had never really questioned it. Apparently, it is only conjectured but widely believed that the decimal expansion in base $10$ of $\pi$ ...
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### Given S, count numbers A such that A - reverseA = S

Given integer $S$ up to $10^9$ count all numbers $A$ such that $A - A' = S \text{ and } A' < A$ and $A'$ is reversed form of $A$ (e.x. $A = 18, A' = 81$). For example if $S = 9$ the answer is $8$, ...
337 views

### Hopping to infinity along a string of digits

Let $s$ be an infinite string of decimal digits, for example: \begin{array}{cccccccccc} s = 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 & \cdots \end{array} Consider ...
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### Replacing a natural number containing a certain digit with the sum of two without that digit

A question in Google Code Jam 2019 qualification round wanted a positive integer n which contains at least one digit 4 to be represented as a sum of two positive integers a and b, neither containing 4....
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### Normal vs disjunctive vs lexicon

Apologies for lack of rigour but I'll attempt to phrase this in an answerable way. In this question, @Charles writes: [Being a normal number] (or even the weaker property of being disjunctive) ...
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### The “special” number $8263$

Prime $8263\equiv 1\pmod {17}$ and $8\cdot 2\cdot 6\cdot 3\equiv -1\pmod {17^2}$. Are there other odd primes $p$ without digit $0$ such that: $p\equiv 1\pmod q$ and the product of the digits of $p$ ...
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### Can we expect infinite many primes $p$ equal to the start of $frac(\sqrt{p})$?

The following routine searches for prime numbers $\ p\$ equal to the beginning of the fractional part of the decimal expansion of $\ \sqrt{p}\$ : ...
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### Hopeless Numbers [closed]

Beatriz Viterbo has called a positive integer which is not divisible by any of the ($2^n$, where $n$ is the number of its digits) numbers that result by introducing a plus or minus sign to the left of ...
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### The number 37 trick - generalization

Suppose we have a number $aaa\ldots a$ composed of $k$ equal digits $a$ in base $b$. Let's divide the number by the sum of its digits. When do we get an integer result? I am reading a book which ...
55 views

### Are there any non-trivial examples of this decimal-binary property

My birthday is 10th of October or 1010 in MMDD format. I just realized that 1010 contains two copies of the number 10 and if spelled out in binary, $1010_2=10$ I was wondering how many other ...
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### Find the digit in hundred-thousandth place of sum

Sum: $1 + 3 + 9/2 + 27/6 + 81/24 + \ldots$ This is a problem on a competitive mathematics test, and I am trying to master the concept so I can understand when similar problems show up in future ...
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### How many even numbers could be formed [closed]

How many even numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, 7 and 8?
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### Questions about 0.999… equals 1 [closed]

Being 0.999... = 1, I expect that they have the same behaviour when applying the same algorithm/operation, but: If we define >, <, =, as checking digit by digit two number, we have that 0 < 1 ...
117 views

### Primes with digits only 1

Let $Y(k)$ be the number consisting of $1$, repeated $k$ times. We know that $Y(2) =11$ is prime. It so happens that $Y(19)$ and $Y(23)$ are also prime. Are there any more? Regards, David
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### Digits in product of two numbers

When we multiply a $m$ digit number with a $n$ digit number, the product will have either $m+n$ digits or $m+n-1$ digits. I want to get some condition on the numbers so that we can predict about these ...
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### Show that decimal addition on real numbers is well defined

I am currently self-studying Hubbard's textbook on Vector Calculus but am stuck on the logic for this question: 0.4.1(a): "Let $x$ and $y$ be two positive reals. Show that $x + y$ is well defined by ...
164 views

### Count the 8-digit integers with 1s and 0s satisfying a digit sum property

This question appeared in a contest in Indonesia in 2011, this is #10. Find the number of positive integers which satisfy the following conditions: It contains 8 digits each of which is 0 or 1. The ...
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### Translating binary plaintext into alphabetic plaintext using an $18$-digit base-$26$ integer system

I'm working on a cryptography problem and went through the long process of decrypting a sent message to get a $27$ digit number. It then says: Plaintext blocks have $18$ letters and that such an ...
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### Sum of digits of $a^b$ equals $ab$

The following conjecture is one I have made today with the aid of computer software. Conjecture: Let $s(\cdot)$ denote the sum of the digits of $\cdot$ in base $10$. Then the only integer ...
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### Why 0.33… is the only expression of 1/3? [duplicate]

I am an undergraduate math student who loves mathematics very much, and I am confused by a math problem. Given $1/3$, we know that $0.33...$ (there are infinite $3$s) is the decimal expression. But ...
With Dirichlet's theorem, we can easily prove that infinite many primes end with a given digitstring with final digit $1,3,7$ or $9$. Can we also prove that infinite many primes begin with a given ...
I was wondering about how we could prove the completeness of $\mathbb R$ when this set is defined to be the set of all decimal expressions of the form : \underbrace{-}_{\text{sign}}\underbrace{317}_{...