Questions tagged [integers]

For questions about the structure, definition, and basic properties of the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$. Do not use this tag just because your question involves integers. Consider using (elementary-number-theory) or (number-theory) instead of or in addition to this tag.

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28 views

Big Matrix Small Determinant

From a programming competition: Construct a square matrix with $N$ rows and $N$ columns consisting of non-negative integers from $0$ to $10^{18}$, such that its determinant is equal to $1$, and there ...
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38 views

Set notation using integers [duplicate]

I am confused with the set notation used with integers, such as $\Bbb Z/n$, $\Bbb Z_n$,and $\Bbb Z/n\Bbb Z$. Please explain the difference between these notations.
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60 views

What is the sum of all integers a such that $a^2-7a-7$ divided by $a-4$ yields an integer?

I sat down for several minutes over the span of a few days to try to solve this problem. I tried different methods, however, the only method I could devise was about making a well educated guess! I ...
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1answer
33 views

Find the set of natural numbers $n \in \mathbb{N}$ for which $\lfloor\sqrt{n+1}\rfloor \neq\lfloor\sqrt{n}\rfloor$

I have to find the set of natural numbers $n \in \mathbb{N}$ for which $\lfloor\sqrt{n+1}\rfloor \neq\lfloor\sqrt{n}\rfloor$. I have tried writing the formal definition of the floor function and tried ...
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34 views

From where this theorem about integer valued polynomials is coming from?

In his article "Polynomials with Integer values" (Resonance), Sury proved an interesting lemma: If $P$ is a non constant, integral valued, polynomial, then the number of prime divisors of ...
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1answer
50 views

How many integer solutions $n$ there are for which $n^3-3\;$ is divisible by $n-3$?

How many integer solutions $n$ there are for which $n^3-3\;$ is divisible by $n-3$? WolframAlpha calculated $16$ integer solutions, but I don't know how did it get the solution. I tried to solve for $...
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1answer
128 views

Solve the equation in positive integers. [closed]

Solve the equation in positive integers for $ x,y $ $$ \frac{47}{\sqrt x} \ + \ \frac{43}{\sqrt y} = \frac{1}{\sqrt {2021}}$$ I tried factoring , squaring , putting integer restrictions but didn't ...
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21 views

Congruence of a power of an integer modulo a relatively prime number

I'm a complete noob in number theory, and I came across this (probably very elementary) statement: if for all $m\in \mathbb Z/n$ with $\gcd(m,n)=1$ there is some integer $k$ such that $$m^k\equiv 1\...
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1answer
47 views

How many subsets $S$ of integer interval $[0,n]$ such that $k \not \in S+S$?

After a bit of experimentation, I thought of the following conjecture: Given any $n \in \mathbb{Z}_{\geq 0}$ and $k \in [0,2n]$, we have $$|\{S : (S \subseteq [0,n]) \land (k \not \in S+S)\}| = 2^{|n-...
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Number theory: Linear combos

If there exist two odd positive integers $a$ and $b$, where $a=\frac{n}{2}+1$ and $b=\frac{n}{2}-1$ and $m\ge n$ where $m$ is another positive integer, all possible values of $m$ can be represented as ...
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517 views

What are the maximum wins a team finishing at Kth position can have among a round robin tournament of N teams? [closed]

This was the Problem Statement. A round-robin tournament is being played among N teams numbered. Every team plays with all other teams exactly once. All games have only two possible results - win or ...
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1answer
42 views

Isn't $\lceil \frac{xy}{z}\rceil =\lfloor \frac{xy+y-1}{z}\rfloor$ for any positive integers $x,y,z$?

I was pretty sure that: $$\left\lceil\frac{xy}{z}\right\rceil = \left\lfloor\frac{xy+y-1}{z}\right\rfloor$$ for positive integers $x,y,z$. But I'm getting wrong results testing it in Python 3: ...
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1answer
38 views

Regularity of $\mathbb Z/ n \mathbb Z$.

For which values $n$ is the ring $\mathbb Z/ n\mathbb Z$ regular? We know that when $n$ is prime, this is regular. When $n$ is not square-free, it's not regular because it is not reduced. However, ...
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3answers
51 views

Divide the first 5n positive integers into $5$ groups such that the sum of the numbers in each group is equal

Prove that for every integer $ n> 1$, it is possible to divide the first 5n positive integers into $5$ groups such that each group has exactly $n$ numbers and the sum of the numbers in each group ...
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1answer
101 views

Showing that it is not possible that for every $q_j$ it holds that $2+\prod_{k \neq j} q_k $ is divisible by $q_j$.

Let $n\ge 1$ and let $Q= \{q_1,\cdot\cdot, q_n\}$ be a set of $n$ odd primes, all different and such that $Q \neq \{3\}$. Show that there is no set $Q$ such that for every $q_j$ it holds that $2+\...
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14 views

Proving well ordering principle with well ordering principle

Hypothesis: "Every nonempty subset S of the positive integers has a least element." Goal: "Every nonempty subset S of non-negative integers has a least element." Is this proof ...
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2answers
51 views

Find the minimum natural number $n$, such that the equation $\lfloor \frac{10^n}{x}\rfloor=1989$ has integer solution $x$

Find the minimum natural number $n$, such that the equation $\lfloor \frac{10^n}{x}\rfloor=1989$ has integer solution $x$. My work- $\frac{10^n}{x}-1<\lfloor \frac{10^n}{x}\rfloor≤\frac{10^n}{x}\...
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1answer
46 views

Working out if a given relation is reflexive, symmetric or transitive (or all 3?) [duplicate]

On the set of integers, let 𝑥 be related to 𝑦 precisely when x ≠ y Is this Reflexive? Is this Symmetric? Is this Transitive? I'm also wondering if it can be multiple? I assume it can maybe be two ...
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Prove that N, Z and Q are discrete sets or do not have the property of continuous:

Prove that natural numbers set N, integer set Z and Rational numbers Q, do not have the the property of continuity. So I have done the proof of R having the property of continuity, but I do not have ...
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1answer
51 views

Prove that $|x-2y|$ is the squaring number with $x^2-4y+1=2(x-2y)(1-2y)$

For $x,y$ are positive integer, statisfying $x^2-4y+1=2(x-2y)(1-2y) \,\,\ (1)$. Prove that $|x-2y|$ is a square number. I try to compute, from $(1)$ We have $(1) \Leftrightarrow x^2+4xy-2x-8y^2+1=0$ ...
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1answer
59 views

How do I randomly choose a number in a smaller range than what my random number generator produces?

My uniform random number generator can only produce integers in the range $0 \leq n \leq 255$. I need to use the output from this random number generator to generate another uniformly random integer ...
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1answer
110 views

binomial identity seemingly illogical and impossible. Is there any way it could be true?

There is binomial expression(s) written as $$\sum_{n\geqslant0}\frac{(-3n+2k-3)n!^2}{2(2n+1)(k-1)!^2(n-k+1)!^2 \binom{2n}{n}}=\begin{cases} 0 & \text{if $k=0$,} \\ -1 & \text{if $k\geqslant1$,...
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0answers
31 views

Is it possible to simplify this binomial expression?

There is binomial expression(s) written as $$\sum_{n>0}\frac{(-3n+2k-3)n!^2}{2(2n+1)(k-1)!^2(n-k+1)!^2 \binom{2n}{n}}=0\; if\; k=0\; or= -1\; if\; k>0$$ which simplifies to $$\sum_{n>0}\frac{(...
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2answers
55 views

Prove that there always exists integer greater than $\sqrt{2n}$ and smaller than $\sqrt{5n}$ if $n$ is an integer greater than 0.

I know that $n \in \mathbb{Z}_+$. I want to prove that there exists such $k \in \mathbb{Z}$ that it is greater than $\sqrt{2n}$ and smaller than $\sqrt{5n}$? I can do it by showing inequalities for $n^...
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1answer
94 views

Can $a^2 - 2^b$ be factored if $b$ is odd?

Let $a$ and $b$ be positive integers. If $b$ is even (i.e. $b=2c$ for some positive integer $c$), then $a^2 - 2^b$ can be factored as $$a^2 - 2^b = a^2 - 2^{2c} = (a + 2^c)(a - 2^c).$$ Edited: (...
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2answers
158 views

Integer Values of $\sum_{k=1}^n k^r . \sum_{q=1}^n \frac{1}{q^r}$

For harmonic numbers $H_n = \sum_{k=1}^n \frac{1}{k}$ we know that this sum is never integer for any $n$. The same is true for generalized Harmonic numbers: the sum $\sum_{k=1}^n\frac{1}{k^r}$ is ...
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1answer
37 views

Integer part problem

If $m,n$ are natural non-zero numbers show that $$[x]+[x+1/n]+[x+2/n]+...+[x+m/n]=[nx]$$ for any real $x$ if and only if $m=n-1$. $[x]$ is the integer part of $x$. I know from the Hermite Identity ...
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1answer
914 views

For every positive integer $a$, find a composite number $n$ such that $n|a^n-a$

Hi guys I got this question in the Algebra section of my Math Test and I am not sure how to solve it. Any help would be regarded. The question is written more clearly below For every positive integer $...
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1answer
181 views

How to find numbers like 174

We know Ramanujan number: $1729 = 1^3+12^3 = 9^3+10^3$ The smallest number expressible as the sum of cubes of two positive integers in two different ways. We also know how to find other Ramanujan ...
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1answer
67 views

Why did we use subtraction symbol to represent negative numbers

Why did we use subtraction symbol to represent negative numbers? We could have just used some other symbol to represent negative number such as .5 .6 .11 And we can add these numbers, subtract these ...
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44 views

How to know if a number is a multiple of other number?

I think the definition of a multiple of a number is: If A is a multiple of B then it's possible to represent A as A= B* X, where X is an Integer. (A and B are also integers) Since X must be an integer ...
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0answers
25 views

How to find the angle of a integer vector of coordinates -1, 0, +1

Is there a simple formula that would give me the angle of a 2D vector with integer coordinates in -1, 0, +1. More precisely, I'm looking for the fastest way to calculate the following function: (1,0) -...
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0answers
43 views

How to show that $(n\mathbb Z+a)(n\mathbb Z+b) = n\mathbb Z + (ab)$?

I am studying the equivalence classes of the relation congruence $\operatorname{mod}n$ on $\mathbb Z$. I define for any $A, B\subseteq\mathbb Z$, \begin{align*} A+B & := \{a+b : a\in A, b\in B\}\...
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23 views

Squarefree Integer

Let $K=Q(θ)$, where $θ = (^n√d)$ for some square free integer $d$. Compute $∆[1, θ, θ^2, ..., θ^n$$^−$$^1$$)]$. Suppose if we let $∂=m+n√d∈Q(√d)$ Then having $∂$ polynomial as zero: $(∂-m)^2=(n√d)^2$ $...
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1answer
30 views

homomorphism of zero divisors

Let $G$ be an abelian group and for each positive integer $n$, define $$ G[n] = \{ g \in G: ng = 0 \}. $$ It is easy to check that if $m$ and $n$ are positive integers such that $m$ divides $n$, then $...
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2answers
56 views

(Dis)Prove $u_n(t)=\displaystyle \left(\sum_{i=1}^t i^n \right) \mod (t+1)$

As I was trying to find a generic formula for the sum of the first $n$ integers to the power of $t$, I found this property that I was able to check from $t = 1$ to $600$ (for 12 periods). Lets define ...
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62 views

Greatest integer not exceeds the sum of recursive sequence

The question is from Pui Ching Invitational Mathematics Competition (2019) My attempt: $x_{n+1} = \frac{5}{\sqrt{x_{n}+8}+\sqrt{x_{n}+3}} $ if $x_{n}> 1$,$x_{n+1}< 1$ if $x_{n}< 1$,$x_{n+1}&...
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1answer
85 views

Numbers greater than $26033514998417$ that can be expressed as a sum of two positive 4th powers in more than 1 way

Are there any numbers greater than $26033514998417$ which can be expressed as a sum of two positive 4th powers in atleast 2 ways ? And if there are any such numbers, then, please include the number(s) ...
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1answer
148 views

The lower bound of $|3^p - 2^q|$ - how to derive from Baker's theorem?

In his blog, Terence Tao discussed the lower bound of $\vert 3^p - 2^q \vert$ in the following corollary. Corollary 4 (Separation between powers of $2$ and powers of $3$) For any positive integers $p$,...
2
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1answer
42 views

Multiplication Law for Order on Integers

I'm using the following definitiosn for addition $+$, multiplication $\cdot$, and the relation $\preceq$ on the set of integers: \begin{align*}\tag{I} [(a,b)]+[(c,d)]&:=[(a+c,b+d)] \\ \tag{II} [(a,...
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4answers
142 views

No. of solutions to a integer-based inequality

Let $p, q$ be positive integers such that $q≤99$. Find number of ordered pairs $(p,q)$ such that $$\frac {2}{5} \ < \frac {p}{q} \ < \ \frac {21}{50}. $$ Here's what I could do: Using the fact ...
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2answers
183 views

The distance between power of 3 and the largest power of 2

For some positive integer $k \gg 1$, the largest bit index of $3^k$ is given by $$m \equiv \lfloor k\log_2 3 \rfloor$$ The distance between $3^k$ and $2^m$ can be written as $$ 3^k - 2^m \equiv a_k \...
2
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1answer
95 views

Can I say the integer solution of this equation is unique?

I want to solve the equation $\frac{y!}{(x-2)!(y-x)!}=\frac{1340!}{659!\times679!}$ where $x,y \in \mathbb{Z}$. I think it is natural to say $(x,y)=(681,1340)$ and $(x,y)=(661,1340)$ are desired ...
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1answer
79 views

If Alice gives Bob $m$ candies, then he'll have $n$ times her candies; if Bob gives Alice $n$ candies, then she'll have $m$ times his candies.

I came up with a seemingly innocent problem of recreational mathematics by myself. It goes likes this. Alice and Bob have some different amount of candies ($>1$ each). If Alice gives Bob $m$ ...
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1answer
42 views

Algorithm for finding subsequence with sum of zero [closed]

I came across an awesome algorithm for determining whither a sequence of integers contains a consecutive subsequence which sums to zero. So for example, in the list $3, -5, -2, 5, -3$ The subsequence $...
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2answers
96 views

When does a parabolic graph have integral x-intercepts and vertex? [closed]

Given a quadratic function $f(x) = ax^2+bx+c$ for integers $a$ and $b$ and $c$, what must be true of $a$ and $b$ and $c$ to guarantee the roots of $f$ and the location and value of the minimum/maximum ...
5
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1answer
77 views

Let $n$,$k$ be positive integers so that $1<k<n-1$. Prove that the binomial coefficient $\binom{n}{k}$ is divisible by two distinct primes

Let $n$,$k$ be positive integers so that $1<k<n-1$. Prove that the binomial coefficient $\binom{n}{k}$ is divisible by two distinct primes. My approach was to look at Pascal's triangle and try ...
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0answers
37 views

Paying laborers with whole coins problem

I'm trying to solve a fair payout problem with weighted rewards and indivisible payment units. Here's the basic idea behind the problem. Imagine an empty ware house intended to store stone bricks. N ...
0
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1answer
71 views

Mathematical expression for quotient

I have a $s$ equation as below, $$ s = \frac{y}{(\frac{y}{x+0.5})+1} $$ However the result under $(\frac{y}{x+0.5})$ should be a $\mathbb{N}$ number even result is in decimal, let say the result ...
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1answer
130 views

Is there an expression for $\sum_{k=0}^{\infty} 2^{-(N^k)}$ in terms of an integer $N>1$? [closed]

I was trying to solve for the limit as $n\rightarrow\infty$: $\displaystyle A_n=\sum_{i=1}^{n} a_i; a_i=\frac{2^{-i}}{i}$ and I landed at the inequality $\frac{N-1}{N}\left(S_{n+1}^{(N)}-a_{1}\right)&...

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