Questions tagged [integers]

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

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2answers
43 views

If $3\mid mn$, then $3\mid m$ or $3\mid n$

I'm currently studying proofs and fundamentals, I'm reading a book by my own and I saw this problem. Theorem Let $m$ and $n$ be integers. If $3\mid mn$, then $3\mid m$ or $3\mid n$. My proof was the ...
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1answer
61 views

Finding general term of sequence satisfying $f(m+n)+f(m-n)=\frac 12(f(2n) + f(2m))$ and $f(1) = 1$

The problem asks to compute $f(2020)$ knowing that $f(1) = 1$ and $f(m+n)+f(m-n)=\dfrac 12(f(2n) + f(2m))$ for integers $m,n$ such that $m>n\ge 1$. My try : I conjectured $f(n) = kn^2$ where $k$...
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1answer
23 views

Solving a system of equations involving the floor function.

I have the following system of equations that I am stuggeling with: $$ax\lfloor y\rfloor=k,by\lfloor x\rfloor=d$$ And I know that $x$ and $y$ are bigger than zero and all the other constants are ...
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0answers
114 views
+100

Bouncing Bullet Problem

This is a problem that was presented to me through the google foobar challenge, but my time has since expired and they had decided that I did not complete the problem. I suspect a possible bug on ...
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1answer
56 views

Is there a geometric proof for distributivity of integer addition/multiplication and other similar properties?

I am aware that commutativity, associativity and distributivity of integer addition and multiplication follow from their standard set theoretic definitions but I am looking for something suitable for ...
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1answer
25 views

How do determine set of exclusions of sum given minimums, maximimums and sets of exclusions of the summed integers?

I'm trying to build a language compiler, but have come across a math-specific problem when implementing addition between integral types. I have n integers ...
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1answer
36 views

Formula or algorithm needed related to integers sequences [closed]

Given a number "X" in such a spiral-like progression of integers in successive layers "n" what formula or algorithm is there to find the layer n in which it lies for very large X? (tip: you might ...
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0answers
21 views

Equality of sums in the context of discrete Fourier analysis

I'm in the context of discrete Fourier analysis and, in particular, trigonometric polynomial interpolation. I cannot understand how to prove the following equality: $$ \frac{1}{2}\left(\hat f(-N/2+\...
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1answer
14 views

Prove the inequality $t!>(n^2)^{t-n^2}$, where $n$ is a positive integer and $t>2n^2$ is also a positive integer.

Just like I stated in the title of the question, I need to somehow prove that $$ t!>(n^2)^{t-n^2} $$ where $n$ is a positive integer and $t>2n^2$ is also a positive integer. I thought about ...
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0answers
43 views

Proof for any positive integers n>7 can be written as the sum of three or fewer squares of positive integers. [closed]

Prove that for any positive integer n>7 can be written as the sum of three or fewer squares of positive integers. How will I go about this. I'm lost.
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1answer
63 views

Alternative proof of $a\times0= 0$

I was trying to find a proof of $a\times0 = 0$ by myself (assuming commutativity, associativity, distributivity, etc) and I came up with $$ a+0=a(1) \implies 1 = \frac{a+0}{a} = \frac aa + \frac 0a = ...
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2answers
80 views

Given that $n^4-4n^3+14n^2-20n+10$ is a perfect square, find all integers n that satisfy the condition

So, I tried solving that by $$n^4-4n^3+14n^2-20n+10=x^2\\10=x^2-a^2, a^2=n^4-4n^3+14n^2-20n+10\\10=(x+a)(x-a)$$ but I couldn't find any integers when I solved it
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1answer
49 views

find integer solutions under square root [duplicate]

I have a equation $y = \sqrt{5x^2+2x+1}$ and I'm trying to generate integer solutions. I've tried Vieta jumping but it failed. So I generated by brute force few solutions and find these: x=2, 15, ...
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2answers
20 views

Analysis- Supremum and infimum

I tried to do this by taking $ X=\{1,3,5,7\}$ and $Y =$ set of all odd natural numbers. In this case the inf$(A)$ is negative infinity. And sup$(A)$ is finite. But is it enough to answer the ...
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1answer
20 views

Integer coefficients of cubic equation imply integer roots

Problem: Let $a,b,c$ be three integers for which the sum $ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$ is integer. Prove that each of the three numbers $ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}...
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1answer
20 views

Integral solution to a³+3ab²=4c³

Please help if integral solutions to this equation exists or not A³+3AB²=4C³ Such that A,B,C are disninct I thought we may be possible to prove solutions exists if and only if A=B=C condition is ...
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3answers
51 views

Proof verification: Showing, through Induction, that a set $S=\mathbb{N}$

Let $S\subseteq \mathbb{N}$ where: (i) $2^k\in S$ for all $k\in \mathbb{N}$; and (ii) for all $k\ge 2$, if $k\in S$, then $k-1\in S$. Prove using induction that $S=\mathbb{N}$. So the base case: If $...
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4answers
38 views

If $xy=ab$ and $x<a\leq b<y$ then $x+y>a+b$.

Let $a,b,x,y$ be positive integers $>0$. Suppose $$ \begin{align} xy&=ab,\\ x<a&\leq b<y \end{align}. $$ Then how to show that $x+y>a+b$? I saw this statement in a comment in the ...
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2answers
30 views

Optimize integer division

I have two positive integers $x$ and $y$. I need to calculate $\frac{100x}{x+y}$ and $\frac{100y}{x+y}$, which sum up to $100$ of course. However, I can only perform integer division. And since the ...
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1answer
20 views

Regular polygon with integral ratio between apothem and side

Let $n$ be an integer. Find at least one $n$ such that the ratio between tha apothem and the side of a regular polygon with $n$ sides is an integer. I found this problem while I was casually playing ...
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1answer
44 views

Variations of random coprime integers probability

The probability for two random integers to be coprime is $\frac{6}{\pi^2}$ (see for example this post), that is about $61\%$. After some computations, for $u_i, v_i$ random integers, the probability ...
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1answer
31 views

Irreducibility of polynomials degree $3$

Let $f(x) = 2x^3+ax^2+bx+c$ where $a,b,c \in \Bbb Z$. Prove that $f$ is irreducible in $\Bbb Q[x]$ if and only if $f(d/2)$ does not equal $0$ for all $d \in \Bbb Z$. I'm not really sure how to start ...
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4answers
32 views

Bijection of the set of natural numbers onto the set of integers. [closed]

An example in Real Analysis by Sherbert and Bartle tells that the set of integers is a bijection of the set of natural numbers. How is the one to one correspondence possible for the set of integers? ...
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1answer
21 views

Find integer values that when multiplied together equal a given value

Given a = bc, with a known integer a, is it possible to find all b and c values that are integers quickly without testing each b and c value? As an example a = 194920496263521028482429080527, is it ...
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0answers
11 views

Identifying a number is an integer, fraction or decimal.

Q- Is $\dfrac{24}{6}$ an integer,fraction,decimal? I think it is an integer and a fraction. Its an integer because $\dfrac{24}{6}$ can be reduced to get an integer. It is not decimal because it doesn'...
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2answers
61 views

Relation between $x^{x+1}$ and $(x+1)^{x}$, $x \in \mathbb{Z}$

So say that we have a pair $(x^{x+1},(x+1)^{x})$ for all $x \in \mathbb{Z}$. Is there any correlation between the members of this pair? Or are they not related?
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0answers
29 views

Sum of real numbers equal to integer number

Let $t_1, t_2, ..., t_n \in \mathbb{N}_{>0}$, $t = \sum_{i=1}^n t_i$, and $s \in \{n,n+1, ..., t\}$. Does an integer function $f: \mathbb{R}_{>0} \to \mathbb{N}_{>0}$ such that $ \sum_{i=1}^{...
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1answer
41 views

Explicitly describing the subgroups of $\mathbb{Z}^{3}$

I am interested in understanding all the subgroups of $\mathbb{Z}^{3}:=\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$. $\mathbb{Z}^{3}$ a free abelian group of rank three, so all subgroups are free ...
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1answer
332 views

Is there a system of mathematics where $4>2$ is false?

A recent question on propositional logic posted on Philosophy Stack Exchange yielded an answer which states, in part, that, The fact that $4$ is greater than $2$ is not a "logical fact" but and [...
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1answer
24 views

Range of values of $k^2$ equal to the sum of two squares and the difference of two other squares

This is an extension of the post linked here: Show that any square number $k^2$ can be written as the sum of two squares and difference of two other squares. Let $k$ be a nonzero positive integer. ...
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1answer
35 views

If $0\leq x\leq1/2$, then why $\inf_{k\in\mathbb{Z}}|x+k|=x$?

Suppose that $0\leq x\leq1/2$. Then how do I formally prove the (rather intuitive) identity $$\inf_{k\in\mathbb{Z}}|x+k|=x?$$ It is easy to see that $\inf_{k\in\mathbb{Z}}|x+k|\leq x$, even without ...
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0answers
45 views

Solution density of a Diophantine equation

In this Numberphile video, Andrew Booker states that: "The is a conjecture density, there is a formula for how may we expect roughly up to a certain number of digits, what is special about $3$ is that ...
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3answers
37 views

Integer solutions to a equation

Let and be positive integers bigger than n. If: Prove that: The first 3 pairs (k,c) and their ratio: (3,2) - 1.5 (17,12) - 1.41666... (99,70) - 1.414285714...
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2answers
25 views

Diophantine equation solving method

So, the equation is $x(x-4)+y(6-y)=10$. Now, I was thinking of checking all pairs of $z+t=10$ if they can be substituted in the equation and get an integer result. Is there a more efficient method?
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1answer
27 views

Why are the integral points on elliptic curve preserved under this substitution?

I have the following elliptic curve: $$3b^2-b=a^3+a^2$$ When I subsitute $\left(a,b\right)=\left(\frac{X-1}{3},\frac{Y+2}{9}\right)$, I get the folloing minimal Weierstrass equation: $$Y^2+Y=X^3-3X+...
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0answers
42 views

Proof that [(1 + √3) ²ⁿ⁺¹] is divisible by 2ⁿ⁺¹ ( [x] denotes the greatest integer function of x) for n >= 0

I came up with a proof but I am not sure if that is correct. I am not sure whether this is rigorous proof, but I think I have a proof for the fact that $[(1 + \sqrt{3})^{2n+1}] = k2^{n+1}$ for $k \in ...
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1answer
29 views

2-split of $n$ is $\left\{ \lfloor \frac{n}{2} \rfloor,\lceil \frac{n}{2} \rceil \right\}$. What about 3, 4, …?

Clarification: $k$-split of $n$ is an ordered integer sequence $\left\{ a_1,\cdots,a_k \right\}\quad \text{s.t.}$ $0\le a_1\le\cdots\le a_k$ $a_1+\cdots+a_k=n$ ${\left(a_k-a_1\right)}$ is minimized. ...
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0answers
34 views

When defining closure property of the set $\{0,1,-1\}$ under addition? Can we say that this set is closed under addition?

When defining closure property of the set $\{0,1,-1\}$ under addition? Can we say that this set is closed under addition? My confusion is that if we take pairs like $(0,1),(0,-1),(1,-1)$ then the ...
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1answer
28 views

How many integers do we need to select from a set from 1 to 20 to guarantee there will be two of the same pairwise sum?

What is the intuition for this? I currently know that there are 38 possible sums, and I'm stuck after that. Any intuition would help, thanks!
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0answers
11 views

Simplest expression of a stair-like sequence with start, end, height of each stair known?

As the title suggests, the start $a_n$, end $b_n$, height $h_n$ of $n^{th}$ stair in a stair-like integer sequence $F_n$ is known, where $F_n=h_i$, iff $a_i\le n\le b_i$, and $F_n$ changes linearly ...
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1answer
34 views

Finding boundaries for a Diophantine equation

I have the following equation: $$k(k+1)(kx+376-x)=376n(nx+2-x)$$ Where $x\in\mathbb{N}$, $x\ge3$, $k\in\mathbb{N}$, $k\ge3$, $n\in\mathbb{N}$ and $n\ge4$. Now, when I want to look for integer ...
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0answers
43 views

Divisors of a multiple of a prime (or any integer)

I have been studying group theory for some time now, and I have noted that quite a few theorems/proofs considering finite groups rely on results from number theory, a branch of mathematics of which I ...
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1answer
49 views

Unexplained points of intersection of $f(x)$ and $f(\sin(\pi x)+x)$

To my knowledge, $$\sin(\pi x)=0\iff x\in\Bbb{Z}$$ and thus $$x=\sin(\pi x)+x\iff x\in\Bbb{Z}$$ . Using Desmos, I plotted two equations: $$f(x)=\Gamma(x+1)$$ $$g(x)=\Gamma((\sin(\pi x)+x)+1)$$ and ...
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5answers
95 views

If $x+y+2xy=83$, find the value of $x+y$.

Let $x$ and $y$ be integers. If $x+y+2xy=83$, find the value of $x+y$. I tried to multiply both sides by $x+y-2xy$ but I could never manage to simplify it. Is there a better way to solve this ...
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2answers
26 views

Integer within certain interval.

I have to show that there is a unique integer within the interval $[a,b]$ where $a = -\frac{3}{2} + \sqrt{\frac{9}{4} + 2(n+1)}$ and $b = -\frac{1}{2} + \sqrt{\frac{1}{4} + 2(n+1)}$ as well as $n \...
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1answer
44 views

Is 5.0 an integer or decimal number?

Is 5.0 an integer or decimal number? I was asked by one of my friends, we got both confused. I said by definition integer contains no or zero decimal part so it should be an integer. But he said that ...
2
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1answer
60 views

Proof explaination - $\sum_{i=1}^{n} \frac{1}{i}$ is not an integer for $n>1$

I was reading a proof to the following fact: for $n>1$, $\sum_{i=1}^{n} \frac{1}{i} \notin \mathbb{Z}$. The proof is as follows: Denote for prime $p$ by $v_p(a)$ the p-adic valuation of $a$. Write ...
3
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0answers
62 views

Solve a Diophantine equation for a specific value of $x$

I have the following problem: Prove that for a free choosable value of $x$, where $x\in\mathbb{N}$ and $x>7$, that the following equation has no positive integer solutions for $(k,n)$, where $k\...
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0answers
20 views

Are there non null four integers such that sum of each two is a perfect square and sum all of them also perfect square?

I have tried to get $4$ integers $a, b,c, d \neq 0$ such that both of $a+b$, $c+d$ and $a+d, b+c$ are perfect square yield to $a+b+c+d$ is a perfect square ? , I have tried to solve the following ...
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1answer
15 views

Contemplating is the Cartesian product of three integer sets is countably infinite or uncountable?

I have been researching the countability of the Cartesian product of three integer sets (which are countably infinite sets). Although I find information on the infinite countability of the cartesian ...

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