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.
2,611
questions
-3
votes
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53
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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 ...
- 327
2
votes
1
answer
187
views
Separating Gamma function in two independent functions: $ \Gamma(n-m) = f(n)g(m) ?$ [closed]
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 $n > m$, ...
- 37
0
votes
0
answers
23
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 ...
- 16k
2
votes
0
answers
66
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 ...
- 31
-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 ...
- 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\...
- 37
8
votes
4
answers
880
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 ...
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 ...
- 10.7k
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
...
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 ...
- 1,691
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 ...
- 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 ...
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$ ...
- 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$$
I want to estimate how many positive integer tuples $(x_1, x_2, \dots x_n)$ that satisfy that inequality. What is the ...
- 6,813
1
vote
0
answers
60
views
A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
2
votes
1
answer
126
views
An inequality with a "Cauchy-Schwarz" flavour
Let $x_1\leq x_2\leq ... \leq x_n$ and $y_1<y_2< ... <y_n$ be positive integers such that $x_1\geq 2$, $x_i < y_i$ and $y_i + 1<y_{i+1}$. Do we have that $$x_ny_n (\sum_{i=1}^n x_i)^2 \...
- 1,110
0
votes
2
answers
128
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 ...
- 37
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0
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59
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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 ...
- 37
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:
$$
f(x) = \frac{m_n}{m_d}x + \frac{b_n}{b_d}
$$$$
m_n, ...
- 13
0
votes
0
answers
11
views
Constant time function to map set on range
So I have a set of positive integers S (like {1, 5, 9, 29, 12}).With this set, I want to construct an algorithm that can, later on, tell me, given any element, its position on the set in constant time ...
- 23
3
votes
1
answer
90
views
Order type of N and Q
Studying linear orderings, I learned two theorems.
Suppose two linearly ordered sets A and B satisfy the following:
(1) countably infinite,
(2) dense, i.e. if x<z then there exists y such that x&...
- 53
0
votes
0
answers
45
views
Software for finding a closed formula from a list of triples of positive integers
Suppose we have a finite list of $n$ triples of positive integer numbers, as:
$$
\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}.\
$$
Is ...
- 151
4
votes
1
answer
149
views
Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers?
Can this proof that
$\sqrt{2}$
is irrational
be rewritten using only integers?
Most proofs that
$\sqrt{2}$ is irrational
start with assuming that
$2=\dfrac{a^2}{b^2}$
and derive a contradiction.
For a ...
- 108k
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votes
0
answers
101
views
Axiomatic characterization of the integer numbers
Peano axioms characterize natural numbers, they use neither sums nor multiplication. Is there an axiomatic characterization for the integer numbers which uses neither sums nor multiplications? I mean, ...
- 194
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votes
2
answers
204
views
Determine the pairs of integers $(x,y)$ that verify the relation: $x^2y^2+2xy+36=3y^2+8x^2$
the question
Determine the pairs of integers $(x,y)$ that verify the relation:
$$x^2y^2+2xy+36=3y^2+8x^2$$
the idea
Fist of all I tried getting everything on the LHS and write it as a product of ...
- 1,305
1
vote
1
answer
76
views
How can different representations of the same integer be equivalent?
I recently read about a way to define the set of integers as the set of all equivalence classes for some equivalence relation $\simeq$ satisfying $(a,b)\simeq(c,d)$ for $(a, b),\;(c,d)\in\mathbb{N}\...
- 454
0
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0
answers
15
views
Decomposing (unimodular) matrix over integers into product of matrices mod d
If I have a $n\times n$ unimodular matrix $A \in \text{GL}(n,\mathbb Z)$, i.e., with elements $A_{ij} \in \mathbb Z$, is there some way to decompose the matrix into a product of matrices $A=A^{(1)}A^{(...
- 409
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vote
2
answers
63
views
Understanding a step in a functional equation
I was trying to solve a functional equation and I while I was pretty close to the answer there is a remaining case that has stumped me for a while now. Here is the problem
Determine all functions $f: ...
- 127
1
vote
0
answers
82
views
Decompositions of symplectic matrices over the integers
Given a symplectic matrix $S \in \text{Sp}(2n,\mathbb Z)$ whereby $S^T\Omega S=\Omega$ with
$$\Omega=\left(\begin{matrix}0&I_n\\-I_n&0\end{matrix}\right)$$
what known decompositions exist such ...
- 409
0
votes
1
answer
48
views
Inequality over positive integers
Let $x_1, x_2, \dots, x_n$ and $y_1, y_2,\dots, y_n$ be two sets of positive integers such that $$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$$ I am ...
- 1,110
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votes
0
answers
106
views
For a discrete moving point, find its first coordinates where it lies in a circle
I'm looking for a method to find the smallest $x$ as a function of $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, $c_2$ and $r$ that satisfies the equation.
$$\left(\lfloor a_{1}+x\cdot\left(b_{1}-a_{1}\right)\...
- 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 ...
user1287101
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 $P$...
- 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. ...
- 2,472
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 $$\left(\sum_{i=...
- 1,110
2
votes
1
answer
110
views
For any $N \in \mathbb{N}$ there exist integers $x,y,z>N$ such that $\{\sqrt{x}\}+\{\sqrt{y}\}=1+\{\sqrt{z}\}$
This problem is from the 2017 admission exam of the SNS (see also page 142 of this pdf for the original text in Italian)
The translation of the exercise is, more or less:
Given $N$ positive integer ...
- 2,472
0
votes
1
answer
72
views
Maximum/minimum of $xyz$
Suppose that $x, y, z$ are positive integers such that
$$xy = 48$$
$$yz = 60$$
How can we maximize and minimize $xyz$?
As suggested in comments,
$$xyz = \frac{48\times 60}{y} = \frac{2880}{y}$$
Which ...
user1284128
1
vote
0
answers
58
views
Minimum of $a+b$
Suppose that $a,b$ are integers such that
$$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}$$
Among the divisors of $72$, only $b\in \{2, 8, ...
user1284128
2
votes
2
answers
70
views
Maximum value of $xy$
Let's suppose that $x, y$ are positive integers such that
$$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{...
user1284128
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 ...
- 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 ...
- 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 ...
- 15
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 ...
- 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 ...
user1281788
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.
$$f_{A000217}(x) = \frac{x}{(1-x)^3}...
- 41
1
vote
1
answer
62
views
Polynomial irreducibility over $\mathbb{Z}_{3}[x]$
Let $f(x) = x^4 + x^2 + x + 1 \in \mathbb{Z}_{3}[x]$
Show that $f$ is irreducible over $\mathbb{Z}_{3}$, then factor $f$ over $K = \frac{\mathbb{Z}_{3}[x]}{(f(x))}$
I first tried to use Eisenstein ...
- 93
1
vote
1
answer
61
views
Every structure preserving map from a ring to a boolean algebra ($+ \mapsto \vee, \cdot \mapsto \wedge$) is induced by a prime element?
Define $(n\mid x) = \begin{cases} 1 \text{ if } n \text{ divides } x \\ 0 \text{ else } \end{cases}$. Let all variables, if untyped be integers. For every prime $p \in \Bbb{Z}$, we have that the ...
- 21.4k
1
vote
2
answers
46
views
$3a$ and $3b-8$ as consecutive numbers
Suppose that $a$ and $b-1$ are consecutive even numbers and $a<b-1$. It seems that $3a$ and $3b-8$ are consecutive as well, although I am unable to show this.
I realize that $|a-(b-1)| = |a-b+1| = ...
user1270647
0
votes
2
answers
177
views
Find integers solutions for which bivariate polynomial with bi-quadratic form: $4x^2y^2-4xy^2+1$, becomes a square number
Could you help me to find all integer $x$ and $y$ for which the bivariate polynomial:
$$4x^2y^2-4xy^2+1$$
is a square number, i.e., it can be expressed as $z^2$ for some integer $z$? From the above, ...
- 5,144
4
votes
2
answers
113
views
Finding numbers that satisfy $AB-BA = 54$
Let's suppose that $AB, BA$ are two-digit numbers. How many different
numbers $AB$ exist that satisfy $AB-BA = 54$?
If we write $AB = 10a + b$ and $BA = 10b+a$, then
$$AB-BA = 54\equiv (10a + b)-(10b+...
user1270647