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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Is there a mathematic formula for n+ceil(n/2)+ceil(n/4)+...

Is there a mathematic formula for n + ceil($\frac{n}{2}$) + ceil($\frac{n}{4}$) + $\dots$? I know that $n + \frac{n}{2} + \frac{n}{4} + ... = 2n - 1$. Currently, I'm calculating each term and adding ...
M.kazem Akhgary's user avatar
2 votes
1 answer
177 views

Separating Gamma function in two independent functions: $ \Gamma(n-m) = f(n)g(m) ?$

I've encountered a problem in my PhD. I would greatly appreciate any suggestions, tips, or comments you might have. The problem is Let $\Gamma(z)$ be the gamma function. Given integers $n > m$, ...
curiosity96's user avatar
0 votes
0 answers
22 views

(sub)monoids of the positive integers under multiplication, with density $0$ in the positive integers, are always multiplicative norms of rings?

Consider integer polynomials of type $"x"$ where we take as imput nonnegative integers. With these nonnegative integer imputs we strictly generate a subset of nonnegative integers ; the set $X$. The ...
mick's user avatar
  • 16k
2 votes
0 answers
62 views

Integer Solutions to the Equation $(n-1)(x+1)(y+1)(z+1)=nxyz-1$

How can I find all integer solutions for the equation $$(n-1)(x+1)(y+1)(z+1)=nxyz-1$$ for any given positive $n$ where $n≤x≤3n-2$ and $x≤y≤z$? All attempts by me to solve this problem have so far come ...
aviolette's user avatar
-1 votes
0 answers
33 views

Existence of solutions in diophantine equations

Recently I've wondered if one equation in integers had a solution and faced a stunning (as for me) question. There is a method of proving that equation doesn't have solutions by looking at it over the ...
cooki's user avatar
  • 1
0 votes
1 answer
42 views

Show that in $\mathbb{Z}/p\mathbb{Z}$ we have: $x^p-x = x(x-1)(x-2) \ldots (x-(p-1))$. [duplicate]

Here's what I have: Using proof by induction we have that when $p=2$ we get $x^2-x=x(x-1)$ which is true in $\mathbb{Z}/2\mathbb{Z}$. Now, assume if $p=k$ then $p=k+1$, i.e., $$x^k-x = x(x-1)\cdots\...
lambdaserb's user avatar
8 votes
4 answers
874 views

Is there a dimensional multiplication operation? [closed]

When expressing numbers with any unit, we know this. We can multiply and divide numbers with different types of units, but we cannot add or compare them. From Terry Tao's 2012 blog post "A ...
Bilgehan Yılmaz's user avatar
1 vote
1 answer
64 views

Integer coefficients in zero linear combinations of dependent vectors

In this answer, Alex Ravsky proves the following lemma: Lemma. Let $K$ and $N$ be positive integers, $V=\{v_1,\dots, v_k\}\subset [0,K]^N$ be a linearly dependent over $\mathbb R$ system of vectors ...
Erel Segal-Halevi's user avatar
1 vote
3 answers
90 views

Find an equivalence relation over all of $\mathbb{Z}$ which has infinitely many equivalence classes with infinitely many elements in each

I want to find an equivalence relation defined on all integers (that is, all of $\mathbb{Z}$) where The equivalence relation partitions $\mathbb{Z}$ into infinitely many equivalence classes; and ...
Christopher Miller's user avatar
1 vote
0 answers
29 views

Asking what is the point of a final claim in Ayres' construction of the set of integers.

Source : Ayres, Outline of Modern Algebra There is a stage in Ayres' construction of the set $I$ of integers that I do not understand. More precisely, I recognize the truth of the claim he makes, but ...
Vince Vickler's user avatar
0 votes
0 answers
19 views

How can one compute rounding preservant integrable functions?

Background & Context : The background of the question is an engineering problem. I want to efficiently represent a set of integers as rounded real valued functions and quickly be able to calculate ...
mathreadler's user avatar
  • 25.9k
1 vote
1 answer
125 views

How do I solve this Diophantine equation?

How do I solve this Diophantine equation, $a^{2n}+b^2=c^{2n}$, where $n$ is any postive integer $>1$ & $a,b,c\ne0$? I tried it by applying the Pythagorean triplets generating formula, but ...
Rajesh Bhowmick 's user avatar
1 vote
2 answers
51 views

Given constrain $m=a_1>a_2>...>a_n$ and the elements are integer prove $\sum \frac{a_i-a_{i+1}}{a_i} < H_m$

For decreasing positive integers $a_1>a_2>...>a_n>0$ when $a_1=m$. Mark $a_{n+1}=0$, Prove that $\sum_{k=1}^n \frac{a_i-a_{i+1}}{a_i} < H_m=\sum_{k=1}^m \frac{1}{k}$ Might add that $n$ ...
Its me's user avatar
  • 593
3 votes
1 answer
83 views

Estimating the number of integer tuples that satisfy a linear inequality

Given a linear inequality in $n$ variables: $$\sum\limits_{i=1}^n c_i x_i \leq b$$ I want to estimate how many positive integer tuples $(x_1, x_2, \dots x_n)$ that satisfy that inequality. What is the ...
Rohit Pandey's user avatar
  • 6,813
1 vote
0 answers
60 views

A question on generalized bases

I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
Dumbest person on earth's user avatar
2 votes
1 answer
125 views

An inequality with a "Cauchy-Schwarz" flavour

Let $x_1\leq x_2\leq ... \leq x_n$ and $y_1<y_2< ... <y_n$ be positive integers such that $x_1\geq 2$, $x_i < y_i$ and $y_i + 1<y_{i+1}$. Do we have that $$x_ny_n (\sum_{i=1}^n x_i)^2 \...
Juan Moreno's user avatar
  • 1,110
0 votes
2 answers
125 views

Roots of $x^2-x+2=0 \in \mathbb{Z}_3[i]$

I've been challenged by a professor to find the roots of $x^2-x+2=0$ in the field $\mathbb{Z}_3[i] = \{a+bi \; \vert \; a,b \in \mathbb{Z}_3\}$. I used the "normal" quadratic formula and got ...
lambdaserb's user avatar
0 votes
0 answers
59 views

Question about modular arithmetic equations

In the book I am reading there is a section about modular arithmetic and modular equations. First, here is what is stated in the book: Theorem: Given the equation $ax \equiv b \mod n$, where $a,b \in ...
NTc5's user avatar
  • 35
0 votes
2 answers
44 views

Finding any arbitrary integer point on a line with rational slope and intercept

Consider the equation of a simple line: $$ f(x)=mx+b $$ with the additional constraint that $m$ and $b$ are guaranteed to be rational numbers: $$ f(x) = \frac{m_n}{m_d}x + \frac{b_n}{b_d} $$$$ m_n, ...
J W's user avatar
  • 13
0 votes
0 answers
11 views

Constant time function to map set on range

So I have a set of positive integers S (like {1, 5, 9, 29, 12}).With this set, I want to construct an algorithm that can, later on, tell me, given any element, its position on the set in constant time ...
pollatron's user avatar
3 votes
1 answer
90 views

Order type of N and Q

Studying linear orderings, I learned two theorems. Suppose two linearly ordered sets A and B satisfy the following: (1) countably infinite, (2) dense, i.e. if x<z then there exists y such that x&...
Lim do's user avatar
  • 53
0 votes
0 answers
45 views

Software for finding a closed formula from a list of triples of positive integers

Suppose we have a finite list of $n$ triples of positive integer numbers, as: $$ \mathcal{L}=\{(a_{i1},a_{i2},a_{i3}):a_{ij}\in \mathbf{N}\setminus\{0\}, \text{ for } j=1,2,3\}_{i=1,\dots,n}.\ $$ Is ...
Hola's user avatar
  • 151
4 votes
1 answer
149 views

Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers?

Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers? Most proofs that $\sqrt{2}$ is irrational start with assuming that $2=\dfrac{a^2}{b^2}$ and derive a contradiction. For a ...
marty cohen's user avatar
0 votes
0 answers
101 views

Axiomatic characterization of the integer numbers

Peano axioms characterize natural numbers, they use neither sums nor multiplication. Is there an axiomatic characterization for the integer numbers which uses neither sums nor multiplications? I mean, ...
RataMágica's user avatar
4 votes
2 answers
204 views

Determine the pairs of integers $(x,y)$ that verify the relation: $x^2y^2+2xy+36=3y^2+8x^2$

the question Determine the pairs of integers $(x,y)$ that verify the relation: $$x^2y^2+2xy+36=3y^2+8x^2$$ the idea Fist of all I tried getting everything on the LHS and write it as a product of ...
IONELA BUCIU's user avatar
  • 1,271
1 vote
1 answer
76 views

How can different representations of the same integer be equivalent?

I recently read about a way to define the set of integers as the set of all equivalence classes for some equivalence relation $\simeq$ satisfying $(a,b)\simeq(c,d)$ for $(a, b),\;(c,d)\in\mathbb{N}\...
naytte2's user avatar
  • 454
0 votes
0 answers
15 views

Decomposing (unimodular) matrix over integers into product of matrices mod d

If I have a $n\times n$ unimodular matrix $A \in \text{GL}(n,\mathbb Z)$, i.e., with elements $A_{ij} \in \mathbb Z$, is there some way to decompose the matrix into a product of matrices $A=A^{(1)}A^{(...
Cameron's user avatar
  • 409
1 vote
2 answers
63 views

Understanding a step in a functional equation

I was trying to solve a functional equation and I while I was pretty close to the answer there is a remaining case that has stumped me for a while now. Here is the problem Determine all functions $f: ...
Ruben's user avatar
  • 127
1 vote
0 answers
82 views

Decompositions of symplectic matrices over the integers

Given a symplectic matrix $S \in \text{Sp}(2n,\mathbb Z)$ whereby $S^T\Omega S=\Omega$ with $$\Omega=\left(\begin{matrix}0&I_n\\-I_n&0\end{matrix}\right)$$ what known decompositions exist such ...
Cameron's user avatar
  • 409
0 votes
1 answer
48 views

Inequality over positive integers

Let $x_1, x_2, \dots, x_n$ and $y_1, y_2,\dots, y_n$ be two sets of positive integers such that $$x_i<y_i \quad \text{for all } i$$ and $$x_iy_i \leq x_{i+1}y_{i+1} \quad \text{for all } i$$ I am ...
Juan Moreno's user avatar
  • 1,110
0 votes
0 answers
106 views

For a discrete moving point, find its first coordinates where it lies in a circle

I'm looking for a method to find the smallest $x$ as a function of $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, $c_2$ and $r$ that satisfies the equation. $$\left(\lfloor a_{1}+x\cdot\left(b_{1}-a_{1}\right)\...
chaise's user avatar
  • 26
1 vote
1 answer
96 views

What is the probability a random integer $x$ when divided by $3$ has a remainder smaller than when $x$ is divided by $9$? without monte-carlo.

I noticed the quantity of numbers from 1-100 with remainder zero modulo nine = quantity of numbers from 1-100 with remainder one modulo nine > quantity of numbers from 1-100 with remainder 2 modulo ...
user avatar
0 votes
2 answers
41 views

Seeking for help, polynomial, periodic sequence, gcd

I am seeking for help about the following exercice : Let $P, Q \in Z[X]$, with no common roots, show that the sequence defined by $u_{n}=gcd(P(n),Q(n))$ is periodic. My Idea is that we can see that $P$...
uvw's user avatar
  • 270
2 votes
2 answers
182 views

The sum of the square roots of $n$ distinct squarefree integers is never an integer. Elementary proof [duplicate]

I'm trying to prove that, if $k_1, \cdots k_n$ are distinct squarefree integers than $$ \sum_{i=1}^n \sqrt{k_i} \notin \mathbb{N} $$ without using the theory of algebraic extensions nor Galois theory. ...
Marco's user avatar
  • 2,447
1 vote
0 answers
42 views

Help proving / disproving an inequality of sums of positive integers

Let $x_1, x_2, \dots, x_n$ and $y_1, y_2,\dots, y_n$ be two sets of positive integers such that $$x_iy_i \leq x_{i+1}y_{i+1} \quad \text{for all } i$$ I need to prove if the inequality $$\left(\sum_{i=...
Juan Moreno's user avatar
  • 1,110
2 votes
1 answer
110 views

For any $N \in \mathbb{N}$ there exist integers $x,y,z>N$ such that $\{\sqrt{x}\}+\{\sqrt{y}\}=1+\{\sqrt{z}\}$

This problem is from the 2017 admission exam of the SNS (see also page 142 of this pdf for the original text in Italian) The translation of the exercise is, more or less: Given $N$ positive integer ...
Marco's user avatar
  • 2,447
0 votes
1 answer
72 views

Maximum/minimum of $xyz$

Suppose that $x, y, z$ are positive integers such that $$xy = 48$$ $$yz = 60$$ How can we maximize and minimize $xyz$? As suggested in comments, $$xyz = \frac{48\times 60}{y} = \frac{2880}{y}$$ Which ...
user avatar
1 vote
0 answers
58 views

Minimum of $a+b$

Suppose that $a,b$ are integers such that $$a^2 b = 72$$ How can we minimize $a+b$? We must require that $b>0$ as $a^2>0$. $$a^2 = \frac{72}{b}$$ Among the divisors of $72$, only $b\in \{2, 8, ...
user avatar
2 votes
2 answers
70 views

Maximum value of $xy$

Let's suppose that $x, y$ are positive integers such that $$x+4y = 40$$ How can we maximize $xy$? I thought of re-writing $xy$ as follows: $$xy = \frac{1}{8}\biggr[(x+4y)^2-(x^2+16y^2)\biggr] = \frac{...
user avatar
2 votes
0 answers
86 views

Functional equation of an integer-valued function

I'm trying to find a function $g$ that satisfies $$g\big(x,\ 2y(1-y)\big)=2g(x,\ y)\big(1-g(x,\ y)\big)$$ for all $x,y∈\mathbb{Z}$ and whose output also lies in $\mathbb{Z}$. Another requirement is ...
Aberone's user avatar
  • 212
1 vote
1 answer
128 views

Find all positive integer $a,b,c$

One day I had a question. When $a≧b$, $a!+b!+1=c^{ab}$ , find all integer. Attempt I would use prime factors to compare each side. $a!+b!+1=b!(a…(b+1)+1)+1$ Comparing each side, $c$’s prime factor is ...
ykk's user avatar
  • 105
0 votes
1 answer
52 views

Formula for number of monotonically decreasing sequences of non-negative integers of given length and sum?

What is a formula for number of monotonically decreasing sequences of non-negative integers of given length and sum? For instance, if length k=3 and sum n=5, then these are the 5 sequences that meet ...
JacobEgner's user avatar
1 vote
1 answer
40 views

Count bounded integers with bound on sum

Given two integers $x$ and $y$, each with lower and upper bounds ($x_{lb} \leq x \leq x_{ub}$ and $y_{lb} \leq y \leq y_{ub}$), count how many pairs have sum between $s_{lb}$ and $s_{ub}$. Of course ...
Stevineon's user avatar
  • 175
2 votes
3 answers
112 views

What does one suppose $\sqrt{x}+\sqrt{y}$ an integer is to solve which of $x,y$, $\sqrt{x}$, $x\cdot y$ are integers?

If the square of $\sqrt{x}+\sqrt{y}$ is an integer, which of the following must be a perfect square, if $x$ and $y$ are positive integers? Choose all that apply. A:$x$ B:$y$ C:$\sqrt{x}$ D:$x+y$ E: $x ...
user avatar
0 votes
0 answers
58 views

Finding the Intersection of two known integer sets

The two sets I am hoping to intersect are the set of triangle numbers and quarter-squares. From those linked resources I have the generating function for both sets. $$f_{A000217}(x) = \frac{x}{(1-x)^3}...
Brandan's user avatar
  • 41
1 vote
1 answer
62 views

Polynomial irreducibility over $\mathbb{Z}_{3}[x]$

Let $f(x) = x^4 + x^2 + x + 1 \in \mathbb{Z}_{3}[x]$ Show that $f$ is irreducible over $\mathbb{Z}_{3}$, then factor $f$ over $K = \frac{\mathbb{Z}_{3}[x]}{(f(x))}$ I first tried to use Eisenstein ...
AANICR's user avatar
  • 93
1 vote
1 answer
61 views

Every structure preserving map from a ring to a boolean algebra ($+ \mapsto \vee, \cdot \mapsto \wedge$) is induced by a prime element?

Define $(n\mid x) = \begin{cases} 1 \text{ if } n \text{ divides } x \\ 0 \text{ else } \end{cases}$. Let all variables, if untyped be integers. For every prime $p \in \Bbb{Z}$, we have that the ...
Daniel Donnelly's user avatar
1 vote
2 answers
46 views

$3a$ and $3b-8$ as consecutive numbers

Suppose that $a$ and $b-1$ are consecutive even numbers and $a<b-1$. It seems that $3a$ and $3b-8$ are consecutive as well, although I am unable to show this. I realize that $|a-(b-1)| = |a-b+1| = ...
user avatar
0 votes
2 answers
177 views

Find integers solutions for which bivariate polynomial with bi-quadratic form: $4x^2y^2-4xy^2+1$, becomes a square number

Could you help me to find all integer $x$ and $y$ for which the bivariate polynomial: $$4x^2y^2-4xy^2+1$$ is a square number, i.e., it can be expressed as $z^2$ for some integer $z$? From the above, ...
Amir's user avatar
  • 4,764
4 votes
2 answers
111 views

Finding numbers that satisfy $AB-BA = 54$

Let's suppose that $AB, BA$ are two-digit numbers. How many different numbers $AB$ exist that satisfy $AB-BA = 54$? If we write $AB = 10a + b$ and $BA = 10b+a$, then $$AB-BA = 54\equiv (10a + b)-(10b+...
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