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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1answer
16 views

Prove GCD statement with multiple variables

For positive integers $a_{1}, a_{2}\cdots, a_{k},$ define $\gcd(a_{1}, a_{2}\cdots, a_{k})$ to be the largest positive integer $d$ such that $d$ divides every $a_{i}$ and any positive integer $c$ that ...
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56 views

Prove that the sequence $ \frac{ (2^n-1)(2^{n-1}-1)…(2^{n-k+1}-1) } { (2^1-1)(2^2-1)…(2^k-1) }$ is an integer

For any given integer $n\ge 1$ and $k \in\{1, 2, \dots, n\},$ $$ F_n(k) = \frac{ (2^n-1)(2^{n-1}-1)...(2^{n-k+1}-1) } { (2^1-1)(2^2-1)...(2^k-1) } $$ For example, if $n = 1,$ then $k \in \{1\}$ and $...
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2answers
95 views

Polynomial such that $e^{2i\pi P(n)} \rightarrow 1$

here is a problem i’ve been having quite a lot of trouble with . Let $P$ be a polynomial such that the sequence $e^{2i\pi P(n)}$ converges to 1 $(i^2=-1).$ Show that $\forall n ,P(n)$ is an integer. ...
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1answer
50 views

$p_{i + k} - p_i \neq \text{const}$ for any $k \geq 1$ where $p_i = i$th prime number.

Let $p_i$ be the $i$th prime number. This should be simple to prove: $$ \forall k \geq 1, c \in \Bbb{Z}, \\ p_{i + k} - p_i \neq c, \\ $$ for some $i \geq 2$. But for example: $$ 11 - 5 = 6 \\ 13 - ...
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1answer
27 views

Struggling with the meaning of discrete.

Even though the set of all integers is infinite, is it still discrete? Also, is a finite set of decimals, such as the following set of $3$ decimals $\{ .1, .2, .3\}$ discrete because it's members are ...
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1answer
25 views

Analytic function which is bijective for integer input

Can we construct a function such that it is bijective (one to one and onto) in positive integer domain and range? i.e. function such that it represents every positive integer uniquely for every ...
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1answer
17 views

How might one prove, by contradiction, that $n$ is prime if $\mathrm{gcd}((n-k),(n-2k))=1$ for for all $k$ s.t. $1\leq k\leq(n-3)/2$?

How might I go about proving this by contradiction. I came about this on my own, but I know it is known. It's rather obvious if one looks at a binary representation of Euclid's Orchard. Does that ...
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2answers
92 views

Is $3(c-a)(c-b)(a+b)$ ever a positive cube for coprime $a,b,c$? [closed]

Are there positive integers $q$ and co-prime $a$, $b$, and $c$ such that: $$3(c-a)(c-b)(a+b)=q^3?$$ Edit: I've been asked to "Please provide additional context, which ideally explains why the ...
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What is the smallest number that when substracted from a dividend of an integer division makes it (the integer divison) exact?

My answer: It all boils down to prove that given a quotient $\frac{D}{d}$ where both $D$ and $d$ are natural numbers. There exists a number $a$ such that there is an integer $c$ for which $$D-a=cd$$ ...
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0answers
54 views

Prove if the following sets are topologies in $\mathbb{Z}$

I've been solving the following problem from the start of my general topology course, and I'd like to check if my answers are correct. The problem is: Study if the following sets of subsets of $\...
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2answers
70 views

Let $x \in \mathbb{N}$. Prove that if $x$ is odd, then $\sqrt{2x}$ is not an integer.

I need help verifying my proof the following theorem: Let $x$ be a natural number. Prove that if $x$ is odd, then $\sqrt{2x}$ is not an integer. The following are the definitions of even and odd ...
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3answers
108 views

Proof that a certain fraction is always an integer

Prove that $$\frac{(n+1)(n)^2(n-1)^2...(n-k+2)^2(n-k+1)}{(k+1)(k)^2(k-1)^2...(2)^2(1)}$$ is or is not an integer for $0\leq k \leq n$, where $k$ and $n$ are integer values. This looks like $\frac{(n+1)...
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1answer
37 views

If $a-b$ divides the difference $f(a)-f(b)$, must $f$ be an integer polynomial? [duplicate]

Background: Suppose $f:\mathbb{Z}\rightarrow\mathbb{Z}$ is an integer polynomial. Then $a-b|f(a)-f(b)$ for distinct integers $a, b$. This can be proven by induction on the number of terms in $f$ as ...
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1answer
37 views

For $\gcd(p,q) = 1$ and $\gcd(r,s) = 1$ show $\gcd(ps,rq) = 1$. [duplicate]

For $\gcd(p.q)=1$ and $\gcd(r,s)=1$ show $\gcd(ps,rq)=1$. I tried via contradiction. Let $\gcd(ps,rq) \neq 1$, so let $d > 1$ a common divisor of $ps$ and $rq$, i.e. $ps = dx$ and $rq = dy$ for ...
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1answer
48 views

Prime numbers of the form $\,\frac{11\cdot 100^{n+1}-9\cdot 10^{n+1}-11}{9}$ [closed]

Are there other prime numbers of the form 12···212···21 after 1222222222221222222222221? Thanks a lot.
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1answer
25 views

Solving multivariate quadratic equations over the integers

I am looking for a method (if it exists) to solve over the integers the following sum of squares equation: $$ x_1^2 + x_2^2+x_3^2 + \cdots + x_n^2 = m,$$ with $m \in \mathbb{N}.$ Someone has any idea ...
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0answers
20 views

How to determine if integer K is a member of an infinite integer series?

Suppose I have an infinite integer series like $Q=4N-3$ where $N = 1,2,3\dots$ In this case the series values Q are $1,5,9,$ etc. Is there a way to determine if an integer $K$ is a member of this ...
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0answers
23 views

Prove that for any $x\in\mathbb{R}$, there exists $k\in\mathbb{Z}$ such that $k\leq x$.

In some earlier work, I proved that any non-empty subset of $\mathbb{R}$ that is bounded below has a greatest lower bound. I now want to prove that for any $x\in\mathbb{R}$, there exists $k\in\mathbb{...
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1answer
31 views

Is there a name for this property/theorem of divisibility in modular arithmetic?

Statement: Given $S = az \pm b$ and $T = ak \pm b$, and $a \neq b$: $$S \mid T \iff k \equiv \pm z \pmod S$$ This holds true only if the three instances of $\pm$ multiply to $+$, i.e., all are $+$ or ...
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1answer
46 views

Some alternating sum of integer part of $\frac{kb}{1722}$.

For every integer $k$ coprime to 1722, how can one compute the sum $\sigma(k)= \sum_{1\leq b\leq 1722, (1,b)=1722} \lfloor\frac{kb}{1722}\rfloor (-1)^{b-\lfloor \frac{b}{2}\rfloor-\lfloor \frac{b}{3}\...
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1answer
42 views

Is it always possible to construct this integer sequence?

Let $A_n$ be a sequence of monotonically increasing, positive integers. How can one construct another sequence $a_n$, again positive integers, such that $$\lim_{n \to \infty} \frac{A_n}{\displaystyle \...
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2answers
31 views

7 digit palindrome problem [closed]

This is my problem: From {1,2,3, ..., 9} How many palindromes of length 7 are there, where each digit can appear at most twice
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2answers
653 views

Seems no answer to this system of integer equations

$a^2 + h^2 = x^2, b^2 + h^2 = y^2, a^2 + b^2 + h^2 = z^2$ $a,b,c,h,x,y,z$ are all positive integers, and also $x, y, z$ are consecutive odd positive integers with increasing order.
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1answer
33 views

Is there a mathematical way of finding b, given a to the power of b and a? [closed]

For example, we have $a^b = 256$ and $a = 2$. Can I find $b = 8$ without trying all possible values of $b$ ?
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0answers
21 views

Polynomial Long Division Algorithm: question on wiki examples [duplicate]

I was told the algorithm on Wiki for polynomial long division works for $\mathbb{Q}, \mathbb{R}, \mathbb{Z}$. Using this algorithm on Wiki I now understand it over $\mathbb{Q}, \mathbb{R}$, however ...
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0answers
39 views

Interesting Integer Problem

Find all solutions $(a,b,c,x,y,z)$ such that $z(ca+b)=cxy$ where $a,b,c,x,y,z$ are integers selected from $[1,9]$, and cannot be repeated. Does anyone have methodical way of solving this equation? ...
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1answer
46 views

How To: Calculate The Σ / A of Pascal's Triangle To Base 111

How would I calculate the sum or area of all the numbers in Pascal's Triangle stopping at row 111? Would I refer to this as the sum? Or the area (A) within the triangle? Also, how would I create a ...
0
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1answer
34 views

Find all units, zero divisors and nilpotent elements of $\mathbb{Z}_4[X]$ and $\mathbb{Z}_6[X]$

I've been solving some problems from my abstract algebra course as training for the final exam, and I want to check if my solution to this one is correct: Describe the units, the nilpotent elements ...
16
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3answers
278 views

What are all the integral solutions of $n!=m(m^2-1)$?

Observe that: $3!=2(2^2-1)$ $4!=3(3^2-1)$ $5!=5(5^2-1)$ $6!=9(9^2-1)$ Question: What are all the integral solutions of $n!=m(m^2-1)$? I guess it is just $(n,m) = (3,2),(4,3),(5,5),(6,9)$, but how to ...
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1answer
76 views

Number theory olympiad problem [closed]

This is a problem I have been tackling recently, but I am unsure how to address it. A positive integer $n$ is good if there exists a set of divisors of $n$ whose members sum to $n$ and include $1$. ...
3
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2answers
121 views

Positive integer solution to $a^3=3b^2-2$

The equation $a^3=3b^2-2$ seem to only have one positive integer solution $(a,b)=(1,1)$, but I am unable to prove that. What I did: Google told me that the elliptic curve theory might help me, but I ...
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1answer
77 views

Can “integer” be used as an adjective?

Although "integer" in Latin means "whole", as far as I know in English it is used as a noun. Personally, I have not seen an integer "number", but only "integers"...
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1answer
31 views

Is it possible to get Sign of an integer nonnegative number using 4 basic operations?

I need to get Sign of an integer nonnegative number using 4 basic math operations (summation, subtraction, multiplication, division). Abs or other functions are not allowed. How can I do this?
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3answers
60 views

$f:\mathbb{N}\to\mathbb{Z}$ which is surjective but not injective

I'm looking for a really simple example of $f:\mathbb{N}\to\mathbb{Z}$ which is surjective but not injective. I thought about: $$ f(x)=\begin{cases} 0 & x=0\\ \frac{x+1}{2} & x\text{ is odd}\\ ...
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0answers
36 views

A basic inequality on integers

Is it true that if $a$ and $b$ are integers and $a \leq b$, then if $m$ is any integer, Is it still true that $a+m \leq b+m$? $a \leq b$ implies that $b-a\geq 0$. Now $(b+m)-(a+m)=b-a\geq 0$ and so $a+...
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1answer
52 views

Integer Part Equations…again

This is a plea for readers solutions! Solve $[20 \, x−3] = [15 \, x + 27]$, $[x]$ = integer part of $x$. I have my solution which is spread over five intervals and can solve any $[a \, x+b] = [c \, x +...
2
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1answer
42 views

Numbers that can be expressed as $ab + a + b$

How many positive integers $n$ less than $100$ can be expressed in the form $ab + a + b,$ where $a,b$ are positive integers? I wasn't quite sure how to approach this problem besides simply finding ...
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2answers
53 views

$(3+x)(2+x) = 0 \pmod 3$ Is Modular Arithmetic like any normal equation?

Let's say we have, for the integer x, this modular arithmetic equation $ (3+x)(2+x) = 0 \pmod 3 $ Is it like any normal equation where you can say either $(3+x) = 0 \pmod 3$ or $(2+x) = 0 \pmod 3$ and ...
1
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1answer
96 views

For which values of integer $k$, does the equation $x^2+y^2+z^2=kxyz$ have positive integer solutions $(x, y, z)$

For which values of integer $k$, does the equation $x^2+y^2+z^2=kxyz$ have positive integer solutions $(x, y, z)$ I immediately thought of saying that from symmetry we have that $x\le y \le z$. Also, $...
2
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1answer
62 views

Find real solution $x$ in order to $x + \sqrt{2020}$ and $\dfrac{5}{x} -\sqrt{2020}$ are integers

$x + \sqrt{2020}$ and $\dfrac{5}{x} - \sqrt{2020}$ are integers $\Rightarrow x + \dfrac{5}{x}$ is an integer $\Rightarrow \dfrac{x^2 + 5}{x}$ is an integer $\Rightarrow x^2 + 5\ \vdots\ x$ $\...
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1answer
42 views

Prove that there are prime numbers such as $p_{i}$ such that $A = p_{1}^{a_{1}}*p_{2}^{a_{2}} * … * p_{k}^{a_{k}} , a_{i}> 0 $ [duplicate]

A and B are natural numbers. Prove that there are prime numbers such as $p_{i}$ such that $A = p_{1}^{a_{1}}*p_{2}^{a_{2}} * ... * p_{k}^{a_{k}} , a_{i}> 0 $ Then we consider $ B = p_{1}^{b_{1}}*...
2
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0answers
28 views

The function field model of the integers and “spillover”

This question was originally posed on MathOverflow. Even with a bounty, it got only a couple of comments and no proper answers, so I thought I'd try it here. In a classic blog post, Tao discusses the ...
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3answers
46 views

Terence Tao's definition of subtraction operation to build integers

I am reading the book Analysis 1 by Terence Tao in which he discussed the construction of integers from natural numbers. The integers are constructed from the natural numbers by the operation $...
3
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1answer
117 views

What is the meaning of “well-defined” and how to prove something is “well-defined” [duplicate]

I'm reading the book Analysis 1 by Terence Tao. There is a concept of well-defined which is mentioned a lot of time in the book but I don't fully understand. For example, in the picture below, in ...
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1answer
70 views

Why is -(-x), the negation of a negation, not the product of two negations? [closed]

I was told that the negation of a negation is not the product of two negations. I understand this to be so because if we prove $-(-x)=x$, we do so using the property of negation: $x+(-x)+(-(-x))=0$, ...
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1answer
87 views

What is the last non-zero digit of $((\dots(((1!)!+2!)!+3!)!+\dots)!+1992!)!$?

What is the last non-zero digit of $((\dots(((1!)!+2!)!+3!)!+\dots)!+1992!)!$? Clarification of the given expression: Let $A_1=(1!)!$ To get $A_2$, we add $2!$ to $A_1$ then we take the factorial of ...
0
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1answer
18 views

$f'\rightarrow 0$ , if $f(n)\rightarrow l (n\in\mathbb{Z})$, then $f(x)\rightarrow l$?

Let $f:(0,\infty)\rightarrow \mathbb{R}$ be differentiable and $f'(t)\rightarrow 0$ as $t\rightarrow \infty$. If $f(n)\rightarrow l$ as $n\rightarrow \infty (n\in \mathbb{Z})$, then $f(t)\rightarrow ...
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1answer
28 views

Creating a random task with $x^2+ax+b=0$ where $x$ is an integer [closed]

I need to create random tasks with $x^2+ax+b=0$ where $x$ is an integer. What rules apply to $a$ and $b$?
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1answer
152 views

How many integers from 1 to 100 can be expressed as the sum of two square numbers? [closed]

Is there any specific formula which I can use to solve this problem? ▪How many integers from 1 to 100 can be expressed as the sum of two square numbers?
3
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1answer
60 views

Possible Values of # of Ordered Factorization of Integers

I was trying to find the number of ordered factorizations of integers into parts $\gt 1$. Starting with $0$, the number of ordered factorizations of $n$ forms this sequence. Interestingly, and in ...

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