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Questions tagged [integers]

For questions about the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$.

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18 views

Integer $p$-norm minimzation

Suppose that, $$ x^*=\underset{x\in\Bbb{Z}^n}{\operatorname{argmin}} \left \{ \Vert{x}\Vert_p \right\},$$ where, $$ \Vert{x}\Vert_p=\left(\sum_{i=1}^{n} \vert{x_i}\vert^p\right)^{1/p}.$$ Can we say ...
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0answers
13 views

$p$-norm is sub-modular

For all $p>1$, $p$-norm is defined as, $$\Vert{x}\Vert_p=\left(\sum_{i=1}^{n} \vert{x_i}\vert^p\right)^{1/p}.$$ In this post, it has been shown that "every $p$-norm function is convex" for $x\in\...
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2answers
76 views

How many tuples of {$a, b, c, …$} satisfy $abc… \leq n$?

Let $n$ and $k$ be positive integers. Let $a, b, c, ...$ be $k$ positive integers such that $abc... \leq n$. How many tuples of {$a, b, c, ...$} satisfy the inequality? Note that the tuples {$a=1, ...
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33 views

Find all the values of the paramater 'a', for which the domain of the function contains only one integer.

Recently I have been studying high school mathematics in Russia. Here they have classification tests that allow you to get into the top universities. In these tests, there are some interesting ...
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1answer
88 views

Solve $x^y + y^x = 499$ for positive integer solution

I'm asked to solve this equation $$ x^y+y^x=499 $$ only positive integer solutions are permitted. First I found the apparent solutions $x=498, y=1$ and $x=1, y=498$. I want to look for a way to solve ...
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1answer
57 views

Does this equation yield only primes?

Interested in solving this equation for $x$: $\exp\Big(\frac{n}{\ln(\pi(x))}\Big)=\pi(x)$ for $n=1,2,3,...$ For $n=1$ up to $n=9,$ I got $x=5,11,13,19,29,37,47,59,73.$ $\pi(x)$ is the prime ...
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2answers
16 views

I have the idea of using the transitive property and/or the the integer combination property. I am stuck tho.

\begin{equation} a, b, \text { and } c \text { are integers. Prove that if } a |(b-1) \text { and } 5 a |(c+2), \text { then } a |(2 b+c) \end{equation}
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2answers
53 views

Elegant Proof that $m | xn \implies \frac{m}{(m,n)} | x$ [duplicate]

I have a proof that shows $m | xn \implies \frac{m}{(m,n)} | x$ which leans heavily on prime factorizations. Is there a more straightforward proof? Edit With this question, I was looking for a proof ...
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33 views

Matrix of integer powers

Is there a name for the square matrix ($j=0...n$, $k=0...n$) $M_{jk} = j^k$ (with special case $M_{00}:=1$) and is there a closed general formula for its inverse? I have stumbled upon this while ...
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51 views

Find $n$ integers from $3n$ ones

$n$ is a positive integer. Is the following statement true? For any $3n$ integers, saying $\{b_1,..,b_{3n}\}$. There exists $n$ of them, saying $\{a_1,..,a_n\}$,so that $\forall$ $1\leq i,j,k\leq n$ ...
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1answer
37 views

Convert a real range to an integer range.

Let's say that we have a set of integers in the range $[1, 4]$. Now, I have a function that will calculate a distance between two vectors, and this function returns a real number in the range $[0, 1]$....
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19 views

How to algebraically mirror a finite subset of integers? [closed]

Let's take the set $S=\{0,1,...8,,9\}$ as an example. By mirroring I mean creating a function $f$ such that $f(x) = x$ is $x \in \{0, 4\}$ and $f(x) = 4 -(x \equiv 5)$ otherwise. The above however ...
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2answers
36 views

With this expression, which values of n gives integer results?

I need to know when would this equation give integer values, I think there might be an easy method I am not aware of, so I am asking here to know if such method/technique is known for finding a ...
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1answer
46 views

Find all positive integer numbers $a_1$ such that $a_n$ is a integer number for all $n \in \mathbb{Z^+}$

Let $m$ be a positive integer number and $\left(a_n\right)$ be a sequence such that $a_1\in \mathbb{Z}^+$ and $$a_{n+1}=\left\{\begin{matrix} a_n^2 +2^m \quad &\text{if} \quad a_n<2^m,\\ \...
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3answers
35 views

How many triangles are there with whole number side?

If $a=29$, and $b=21$, how many triangles are there such that side $c$ is a whole number? My approach: Tried using certain equations to establish relationship between sides to maybe point to right ...
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2answers
62 views

Non-collinear integral coordinates in a plane.

Given that A, B, and C are noncollinear points in the plane with integer coordinates such that the distances AB, AC, and BC are integers, what is the smallest possible value of AB? What's the best ...
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3answers
115 views

Show that $a^3 - b^3 = c! - 18$ does not have a solution

Let $a, b,$ and $c$ be positive integers and $c \gt 6$. Show that the equation $$a^3 - b^3 = c! - 18$$ does not have a solution for all positive integers $a, b,$ and $c$. What I have realized ...
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1answer
36 views

Can even/odd classes be applied in triples, quadruples, etc., and have any uses?

The classes of even and odd numbers has many uses, and we can find rules about combining them. An odd number added to another odd number always yields an even number. Even + even = even. Odd + even = ...
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0answers
24 views

Period of a particular Cycle for a Bessel Function

The Bessel Function of the First Kind $J_a(x)$, and the Bessel Function of the Second Kind $Y_a(x)$, at least when $a$, is an integer or half integer are cyclical, as their values go from positive to ...
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1answer
23 views

Find the recursive definition for these two sets of numbers

Good afternoon, I'm struggling with finding the proper recursive definitions for these two sets of numbers: problem 1: 1, –1, 2, –2, 4, –4, 8, –8, 16, –16, … problem 2: 1, 2, 3, 6, 11, 20, 37, 68, ...
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2answers
59 views

$a^{m+n} = a^m*a^n$ where $m,n \in \mathbb{Z}$

I can use the exponents laws only $m,n \in \mathbb{N}$, and need to prove them for $m,n \in \mathbb{Z}$. note that $0 \neq a \in \mathbb{R}$ I proved some cases (mainly the trivial ones) and I'm ...
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3answers
78 views

Is the following result true? Or Is there any known result about fractions like this?

Is the following result true? Or Is there any known result of fractions like this? Let $n$ be fixed. There are infinitely many integer solutions for $$\sum_{i=1}^n \frac{1}{x_i} = 0,$$ where $x_i \...
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1answer
37 views

Power of $2$ that divides $\lceil(3+\sqrt5)^{2n}\rceil$ [closed]

$\lceil(3+\sqrt5)^{2n}\rceil$ is divisible by A. $2^{n+1}$ B. $2^n$ C. $2^{n-1}$ D. not divisible by $2$ original image
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0answers
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Proof of $\sum_{i=0}^{N_s - 1} e^{-j2\pi n(k'-k)/N_s} = N_s\delta(k'-k)$

Currently, I'm faced with this problem: Prove that this property holds when $N_s$ is an even integer, for any integers $n$, $k'$, $k$: $\sum_{i=0}^{N_s - 1} e^{-j2\pi n(k'-k)/N_s} = N_s\delta(k'-...
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5answers
57 views

Prove that for any integer $a$, $9\nmid(a^2-3)$.

I'd like to know if my proof is a valid way of proving this and additionally if there is a better way of going about this? I'm relatively new to discrete mathematics so any critique would be very ...
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2answers
52 views

Finding an equivalence classes for the relation $(x_1,y_1)\sim(x_2,y_2)\ \iff\ x_1^2+y_1^2=x_2^2+y_2^2$.

Let R be the relation defined on the set of integer pairs by $(x_1,y_1)R(x_2,y_2)$ when $x_{1}^{2}$ + $y_{1}^{2}$ = $x_{2}^{2}$ + $y_{2}^{2}$. Prove that R is an equivalence relation and determine ...
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2answers
39 views

Help with mathematical induction! [closed]

My teacher is asking me to: Prove that 2 is a factor of (n+1)(n+2) for all positive integers n. I need to solve this problem through mathematical induction but I am completely lost! Please help me and ...
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3answers
81 views

Don't understand way algorithm is described in paper about Monte Carlo algorithm

I'm reading An Improved Monte Carlo Factorization Algorithm by Richard P. Brent. I'm not sure I'm understanding some of the notation correctly. To my understanding $x_0$ is an arbitrary starting ...
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3answers
37 views

Congruence Class of mod p where p is a prime number

If $n =p$ is a prime number and if $[0]\neq[a]$ is in $J_p$ , then there is an element $[b]$ in $J_p$ such that $[a][b] = [1]$. ($J_p$ being the set of congruence classes mod $p$) This was a remark ...
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0answers
29 views

Divisor Properties

This is a general question regarding divisibility If $c|a$ and $c|b$, does $c|a+b$? Where $a,b,c$ are integers If yes, does this also hold for general rings?
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1answer
74 views

How to calculate the integer number of vertices in a 2 dimensional triangle?

Imagine a 2-dimensional right triangle drawn on graph paper (a lattice), with the right corner originating at (0,0). Each unit on graph paper has a width of 1 unit. The lengths of the base and ...
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Determine the set of integers that are represented by the binary quadratic form (1,0,-1) [duplicate]

I need help with finding the set of integers represented by the form (1,0,-1). This is essentially f(x,y) = x^2 - y^2 which can be factorised into (x + y)(x - y) and the determinant is d = 4 > 0.
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0answers
11 views

Given L, s and d, which are positive real number, what is the probability that there exist integer k and k', such that $kd\in [k'(L+s)-s, k'(L+s)]$

Given $L$, $s$ and $d$, which are positive real numbers, is there always a pair of integers k and k', such that $kd\in [k'(L+s)-s, k'(L+s)]$. It is like there is a line which is painted red of length $...
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1answer
81 views

For every $b$ in the power $a^{b}$, does there exist an $a$ such that the digit sum of this power is equal to $a$?

$1^0 = 1\to 1 =1$ $x^1=x\to x=x\;\forall x$. $9^2 = 81\to 8+1=9$ $8^3=512\to 5+1+2=8$. $7^4=2401\to 2+4+0+1=7$ $46^5 = 205962976\to 2+0+5+9+6+2+9+7+6=46$ $64^6 = 68719476736\to 6+8+7+1+9+4+7+6+7+...
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4answers
387 views

What is the value of $\frac11+\frac13-\frac15-\frac17+\frac19+\frac1{11}-\dots$?

The series $\sum_{k=1}^{\infty }\frac{(-1)^{k+1}}{2k-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$ converges to $\frac{\pi}{4}$. Here, the sign alternates every term. The series $\...
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0answers
27 views

Rewriting a set of integers to get rid of repetition but keeping subset sum ordering

Say, I have a set of 6 +ve integers sorted in ascending order: $A = \{2,4,4,4,5,7\}$ Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them: $\therefore B= ...
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0answers
36 views

Simple Proof of Binomial Theorem for Negative Integer Powers

There's a vast amount of clutter on the internet about this which I've been trawling through but it does not answer exactly what I'm asking which, because superficially similar questions have been on ...
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1answer
57 views

How to show that the following function from $\mathbb{Z} \to \mathbb{Z}$ is injective and/or surjective, or neither?

I have a question that says the following: Define $$f(n) = \frac{n}{2} + \frac{1-(-1)^n}{4}$$ for all $n \in \mathbb{Z}$. Thus, $ f\colon \mathbb{Z} \to \mathbb{Z}$ , where $\mathbb{Z}$ is the set ...
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1answer
49 views

Generators for $\mathbb{Z_n}$

I would like to show that $K$ is a generator for $\mathbb{Z}_n$ $\iff$ $\gcd(K,n)=1$ and $1 \leq K <n$. My Attempt: Assume $\gcd(K,n)=1$ and $1 \leq K <n$. That means $K \in \mathbb{Z}_n$ and ...
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1answer
31 views

2 distinct integers between 5 and 17 inclusive are chosen. What is the probability that their product is odd?

"Suppose two distinct integers are chosen from between 5 and 17 inclusive. What is the probability that their product is odd?" I can't figure out the probability, although I do know that both ...
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3answers
100 views

Inverse of an Integer Matrix

I found a problem on the Open Problem Garden which asks about the conditions on a rectangular, full-rank, integer matrix such that its right inverse (given by: $A^T (AA^T)^{-1}$ ) is also an integer ...
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0answers
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Subgroups of the integers.

Theorem: The subgroups of $\mathbb{Z}$ are $n \mathbb{Z}$ for $n=0,1,2....$. My Proof: Let $H$ be an arbitrary subgroup of $\mathbb{Z}$. Let $x \in H$. If $x<0$ then since $H$ is closed under ...
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2answers
30 views

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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1answer
35 views

Near integers in powers of binomials with radicals

This question comes out of a mathematics calendar problem that asked for the tenths digit of the expression $(17 + \sqrt{280})^{17}$. The calendar implied the digit should be 9, but after playing ...
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3answers
73 views

If f = (x/y) + (y/x) +1/(xy) is an integer. Prove that f must be of the form 3x [closed]

I have tried using Induction method but I am unable to resolve it to a single variable. Also, $x$ & $y$ are positive integers. $f\:=\:\frac{x}{y}+\frac{y}{x}+\frac{1}{xy}$ Edit : This one is ...
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0answers
26 views

Division with replacement of floating-point arithmetic to integer arithmetic

The issues: Not all the hardware has an FPU => not possible to use floating-point arithmetic. Not all the hardware has an uint64_t => not possible to use ...
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1answer
47 views

Does there exist this particular system of subsets of the integers?

Is there a system of subsets of the integers, $\mathcal{E} \subseteq \mathcal{P}(\mathbb{Z})$, such that (B1) $A \in \mathcal{E} \Leftrightarrow A^c \notin \mathcal{E}$ (B2) $A \in \mathcal{E}, A\...
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1answer
39 views

Solving coupled modular equations over the integers with general coefficients

I have encountered a problem in my research that requires solving two coupled modular equations for integers x,y for general integral coefficients. As someone without much experience in discrete math, ...
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1answer
18 views

Is there an ordering of integers with maximum near equal spacing between integers of the same set bit count?

Let $n$ be a whole number, and $\mathcal{S}_n$ be an ordered list of integers from 0 to $2^{n}-1$, does there exist an ordering $\mathcal{D}_n$ of $\mathcal{S}_n$ such that the distance between ...
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1answer
82 views

Prove if there exists $[b]\in{Z}_{pq}$ such that $[b]^2=[a]$ in ${Z}_{pq}$, then there are exactly four distinct $[x]\in{Z}_{pq}$

Let $p$ and $q$ be distinct odd primes. Let $a\in{Z}$ with $gcd(a,pq)=1$. Prove that if there exists $[b]\in{Z}_{pq}$ such that $[b]^2=[a]$ in ${Z}_{pq}$, then there are exactly four distinct $[x]\in{...