Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, radicals, etc.

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14
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552 views

A strange little number - 6174.

Take a 4 digit number such that it isn't made out the same digit (1111, 2222, .. etc) Define an operation on such a four digit number by taking the largest number that can be constructed out of these ...
7
votes
0answers
122 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
6
votes
0answers
107 views

Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
6
votes
0answers
215 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
6
votes
0answers
99 views

Non-standard proofs of standard theorems

In Richard Kaye's book Models of Peano arithmetic, one can read (page 13): We have proved that any nonstandard $M \models \mathrm{Th}(\mathbb{N})$ has a nonstandard $a \in M \models \theta(a)$ iff ...
4
votes
0answers
70 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
4
votes
0answers
86 views

Quotition versus partition

This new Wikipedia article may look different by the time the reader of this question sees it. For now, it says $6\div 2$ can be construed in either of two ways: "How many parts of size $2$ must be ...
4
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0answers
80 views

Regarding identities with sums of consecutive squares

This comic http://abstrusegoose.com/63 points out an interesting identity with sums of consecutive squares. Let us take positive integers $k, p, q$, with $p < q$, and ask if $k^2 + \ldots + (k + ...
3
votes
0answers
30 views

question in product

can any expert just check my solution You bought a car for $\$2500$ down and made payments of $\$299.50$ each month for $36$ months. (a) Find the amount of the payments over the $36$ months. (b) Find ...
3
votes
0answers
55 views

Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { ...
3
votes
0answers
48 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
3
votes
0answers
135 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
3
votes
0answers
35 views

How to find minimum$(|a_n|+|a_{n-1}|+\cdots+|a_0|)$ if $f(x)=0$ has at least one root $x\in (0,r)$?

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0,a_i\in\mathbb Z,n\in\mathbb N.$ How to find minimum$(|a_n|+|a_{n-1}|+\cdots+|a_0|)$ if $f(x)=0$ has at least one root $x\in (0,r)$? (For example, ...
3
votes
0answers
108 views

Do expressions like $(-1)^{2/3}$ show up naturally in pure or applied math?

Let $x$ denote an arbitrary real number. Then $x^n$ makes sense for arbitrary $n \in \mathbb{N},$ via the obvious recursive definition. We can extend this definition by asserting that if $x$ is ...
3
votes
0answers
140 views

Can this be only solved by trial and error?

The following question was asked in a competitive exam Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the ...
3
votes
0answers
96 views

Square and reverse reading of an 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}= ...
3
votes
0answers
181 views

How does my $10\times10$ abacus work?

How does this $10\times10$ abacus work? More specific: Counting: is it common to count from 1 to 100, or from 1 to 9.999.999? How does addition work? How does multiplication work? Are there any ...
3
votes
0answers
220 views

Arithmetic mean sum

Let $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^ng(k)=A $$ Then for what functions $f(x)$ does $$\lim_{n\to\infty}\frac{\sum_{k=1}^n f(k)g(k)}{\sum_{k=1}^nf(k)}=A$$
3
votes
0answers
100 views

Least characters in a numerical representation of integers

I was wondering what the shortest way to represent any given number is. For example, $387420489=9^9$. So, for this case, the smallest representation is of order 2 (2 numbers). Alternatively, ...
3
votes
0answers
1k views

Is there an equivalent to the distributive law for division over subtraction and/or addition?

I understand that the the distributive law cannot be applied to division over addition/subtraction, but is there an equivalent law to expand it out. For example, I know: $$100 \times (5 + 3) = (100 ...
2
votes
0answers
29 views

Interval arithmetic for open intervals

I found a detailed paper which outlines the rules of interval arithmetic for closed intervals, including unbounded closed intervals, but it makes no mention whatsoever about open intervals. I'm ...
2
votes
0answers
155 views

Relationship between two elements of two matrices with two numbers are not elements of the two matrices

I have two matrices, $$A= \left[ \begin{matrix} 5 & 10 & 15 & \cdots \\ 17 & 28 & 39 & \cdots \\ 35 & 52 & 69 & \cdots \\ \vdots & \vdots & \vdots & ...
2
votes
0answers
126 views

The set of all natural numbers is closed under addition

I'm trying to prove the theorem described in the title, but my proof is so obvious I doubt it is sufficient. Here's my way of proving it: Definition of addition: Let a, b, and c be natural numbers. ...
2
votes
0answers
102 views

Inserting +/- into 123456789…

I'm looking at a generalization of the problem of inserting + and/or - into the blocks $123456789$ and $987654321$ to create a formula for $100$, like this: $$123 - 45 - 67 + 89 = 100$$ $$9 - 8 + 7 ...
2
votes
0answers
72 views

Prove $\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 + b^2}\right) \ge 9$

If $a,b,c \in \mathbb{R^+}$,then prove that the following inequality holds: $$\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 ...
2
votes
0answers
39 views

A problem on generalization of a solution

Consider the following question: Two cars A and B start simultaneously from two different cities P and Q respectively and move back and forth between the cities.(As soon as car A reaches city Q ...
2
votes
0answers
87 views

Necessary criterion for a field extension to be normal

I'm working on a lemma concerning some Galois theory and arithmetics. Let $p$ be an odd prime and $K/F$ be a finite Galois extension of number fields of order prime to $p$ with Galois group $H$. Let ...
2
votes
0answers
168 views

Number of solutions for $x$ of such form that $\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$

Consider $$\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$$ where $y$ is an integer. In relation to solutions for $x$; How could one prove that: $(1)$: There are $y$ solutions for $x$, in total. ...
2
votes
0answers
131 views

Is there a finite number of solutions to $\mathrm{Re}(a^n)+\mathrm{Im}(a^n)=b^n$, where $a$ is a Gaussian integer and $b \in \Bbb Z$?

Let $$E_n=\{(x,y,b) \in \mathbb{Z}^*\times \mathbb{Z}^*\times \mathbb{Z}^* ~|~ \gcd(x,y,b)=1 ~ \mathrm{and}~\mathrm{Re}((x+iy)^n)+\mathrm{Im}((x+iy)^n)=b^n \}$$ For $n \geq 3$, is $E_n$ finite or not ...
2
votes
0answers
87 views

Clarification of variable values in Arithmetic Coding algorithm

I have been trying to follow this video to implement my own Arithmetic Coding algorithm in Java. I am having a bit of trouble figuring out what some of the variables in the video should be. For ...
2
votes
0answers
71 views

Need to determine the formula to work out a reduction percentage

We currently calculate a value for a business function as follows; Inputs: A list of values such as; 12 12 6 12 We then add 1 to each of these values and we have values as follows; 13 13 7 13 ...
2
votes
0answers
52 views

How to prove that $G_3>0$ in this case?

Let $\Lambda=\{a+be^{2\pi i/3}|a,b\in Z\}$, then $G_{3}(\Lambda)=\sum_{\omega\in\Lambda-\{0\}}\frac{1}{\omega^{6}}$ should be real and nonzero, but how can one prove that it's positive? Moreover, in ...
2
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0answers
15 views

Decomposability in a context of size constraints on intervals

Let $F$ be a finite set of pairs of positive integers. Say that a set $A \subseteq {\mathbb Z}$ is $F$-admissible iff its intersection with any integer interval of length $a$ has cardinality at most ...
2
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0answers
93 views

Is there a term used to describe both an equation and inequality?

Is there a term used to describe both an equation and inequality? The closest thing I can think of is "relation".
2
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0answers
127 views

1/3+2/3 in double precision

When I add 1/3 and 2/3 in double precision, I ended up with $1.\boxed{111\ldots1}1\times2^{-1}$, where the boxed part is the 52-bit mantissa. By the rounding to even rule, I should round it up, right? ...
1
vote
0answers
71 views

Why does base*height work?

I want to rigorously prove the idea that Base*Height=Area works (I do realise there are shapes which do not satisfy this equation). I think I can see why it works for integer values, but I want ...
1
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0answers
50 views

Given $m^k\le n <m^{k+1}$ find $x$ and $y$ such that $x\cdot m^k+y=n$

Let $n,m,k\in\mathbb{N}$. Assume $m^k\le n <m^{k+1}$. Find $x,y\in\mathbb{N}$ such that (1) $x\cdot m^k+y=n$ (2) $0<x<m$ (3) $0\le y<m^k$ My question: does there exist a general ...
1
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0answers
30 views

Decomposition of a real number in base $b$

We've had a class today about decomposing a real number in base $b$ and saw the following theorem : Let $b\in\mathbb{N},b\geq2$. For all $x\in\mathbb{R}_+$, there exists a unique sequence ...
1
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0answers
35 views

Terminology: Name for general, non-differential equations over $\mathbb{R}^n$

Let $x \in \mathbb{R}^n$ be a variable, and $f_1, \ldots , f_n : \mathbb{R}^n \mapsto \mathbb{R}$ be a sequence of functions, each having a closed form expression. Assume that we want to solve the ...
1
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0answers
21 views

Determine an arithmetic relation

Let $f$ be an arithmetic function. Let $p$ be a prime number, $\chi(n)=\left(\frac{n}{p}\right)$ be a primitive Dirichlet character modulo p, where here $~\left(\frac{n}{p}\right)$ is the ...
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0answers
26 views

Arithmetic and Algebra exercises on latex source code.

I´m currently writing a little book for two student that I teach. The book covers school arithmetics and algebra, and it include theory and examples. Since I don´t have time to prepare a good sets of ...
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0answers
41 views

Finding possible numbers to yield a certain arithmetic mean (average)

I'd like to know how I could find the possible numbers to yield a certain average number (arithmetic mean), given that: the arithmetic mean will be calculated from 20 integer numbers, each having a ...
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0answers
19 views

Help with Calculating amount in pages by 2 working days

I need help with the following, I think I got this right, but wanted to pass it by the gurus/experts first: ...
1
vote
0answers
61 views

Why doesn't this base 10 number x mod 2^y work for converting base 10 to binary

Okay I tried to convert 1 million to binary by dividing by a power of 2 and taking the remainder and dividing that by a power of 2 and so on and I got this: 1111010000100100000 Google says 1 million ...
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0answers
134 views

Time-invariant Differential Equation

I was wondering if it is safe to assume that $$ (1/t)y'(t) + (1/t)y(t) = 0 $$ is a time-varying differential equation. I would say these (1/t) factors cancel out and make the eq. time invariant. ...
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0answers
69 views

A question from Tom Korner's 'The Pleasures of Counting'

Merchant ships sailing independently take 75% of the time to complete voyages compared to ships sailing in a convoy but lose 14% of their number to submarines on each voyage, whilst convoyed ships ...
1
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0answers
41 views

Is undecidability of arithmetic a corollary of Tarski undefinability theorem?

Arithmetic is undecidable, in other words the set of Godel numbers of theorems of arithmetic is not recursive, and so there is no algorithm/ recursive function to decide if a statement is provable or ...
1
vote
0answers
261 views

Property of arithmetic means?

$a,b,c,d \geq 0.$ It seems to me that this inequality is true and equality holds when $a=b=c=d$? $$\dfrac{a+b}{2}\dfrac{b+c}{2}\dfrac{c+d}{2}\dfrac{d+a}{2}\leq ...
1
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0answers
40 views

Divisors and cyclotomic polynomials

Let $n \in \mathbb{N}^{\ast}$ and $\Phi_{n}(X)$ be the $n$-th cyclotomic polynomial defined by : $$ \Phi_{n}(X) = \prod \limits_{\substack{1 \leq k \leq n-1 \\ \gcd(k,n)=1}} \Big( X - \exp \big( ...
1
vote
0answers
30 views

Is $x_1^{\alpha_1} + \dotsb + x_n^{\alpha_n}\geq x_1^{h/n}\dotsb x_n^{h/n}$ an example of power means?

I learned here that there is a relation between weighted means of the form $x_1^{\lambda_1}\dotsb x_n^{\lambda_n}$ and $(\lambda_1 x_1^r + \dotsb + \lambda_nx_n^r)^{1/r}$, namely that the former is ...