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.
2,472
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Is this way of deriving the formula for the number of decimal digits for positive integers correct or justified?
For a positive integer $x$ expressed in decimals, we can write
$$
x = \sum_{k=0}^n a_k \cdot 10^k \tag{$a_k \in \{0, 1, \dots, 9\} \land a_n \neq 0$}
$$
and I want to find a function $d\colon \mathbb{...
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Theorem 2, Section 1.3 of Hungerford’s Abstract Algebra
Every infinite cyclic group is isomorphic to the additive group $\Bbb{Z}$ and every finite cyclic group of order $m$ is isomorphic to the additive group $\Bbb{Z}_m$.
Proof: If $G=\langle a\rangle$ is ...
- 3,188
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How to prove that, among any 𝑛+1 distinct odd integers from {1,…,3𝑛}, at least one will divide another?
This was one of the exercises in my textbook and I've been working on it for well over 10 hours over the span of 3 days without much progress. I don't think that it's even supposed to be a hard ...
- 317
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0
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50
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When is $r^n+p\mathbb{Z}=(r+p\mathbb{Z})^n$?
Given $r, p \in \mathbb{Z}$ and $n\in \mathbb{Z}_{\ge 1}$.
Set $r^n+p\mathbb{Z}:=\{r^n+pm:m\in \mathbb{Z}\}$ and $(r+p\mathbb{Z})^n=\{\prod_{i=1}^n(r+pm_i): m_1,\dots, m_n\in \mathbb{Z}\}$.
Clearly, $(...
- 423
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prove that $x^2-2y^2=1$ has infinite integer solutions. [duplicate]
Prove that $$x^2-2y^2=1$$ has infinite integer solutions.
I know this is a case of n=2 in Pell's equation.
But the thing is we are not allowed to use that as a reference in this problem.
And even when ...
- 649
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5
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Prove that $x+y = (x-y)^2$ has infinite integer solutions
Prove that $$x+y = (x-y)^2$$ has infinite integer solutions.
I tried to reform the equation in several ways.
As $$(x-y)(x-y-1)=2y$$
Or $$(x+y)(x+y-1)=4xy$$
I was trying to find y in terms of x
But as ...
- 649
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0
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25
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Convergent sequence of a special topology on $\mathbb{Z}$
Can we characterize all the convergent sequences of the topology on $\mathbb{Z}$ (Furstenberg's topology) generated by the basic open sets of the form $B(c,r)=\{c+rk \mid k\in \mathbb{Z} \}$, for $c,r\...
- 121
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55
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Number of digits in a computation involving integers with different number of digits
Compute $\max(A+B, C+D) + E$ where $A,C$ have $n$ digits, $B,D$ have $3n$ digits, $E$ has $2n$ digits
What will be the primitive operations?
What will be the output digit?
Hi for the above question I ...
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29
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Find $(a,b)\in\mathbb{Z}^2,\;b^3-2a^3=1$ in a simple way [duplicate]
By reading a proof, the goal was to find all the $(a,b)\in\mathbb{Z}^2,\;b^3-2a^3=1$.
The author then says in one line that "by doing some rough inequalities", we have $(a,b)\in\{(0,1),(-1,-...
- 255
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47
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Formalizing daily use of integer representation
In daily life, it is common practice to use a sequence of number elements as an integer, e.g., 999 is a decimal number. As I am reading rigorous construction of numbers from Rudin's mathematical ...
- 1,748
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76
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Troubles with proving R is an equivalence relation [duplicate]
With $A = \mathbb{Z}$ and $B = \mathbb{Z}-\{0\}$, I'm trying to prove that the relation $R$ defined on $A\times B\,$ by $(a,b)R(c,d)\,$ iff $\,ad=bc$, is an equivalent relations.
While I do see why ...
- 69
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How many integer numbers less than $10^4$, with digits $\{1,2,3,4\}$ only and constraints
How many integer numbers less than $10^4$, with digits $\{1,2,3,4\}$ only and no couples of digits like $\{13, 31, 24, 42\}$?
I calculated all the integers made of $\{1,2,3,4\}$ digits and less than $...
1
vote
0
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47
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Generalization of "Choose" function [duplicate]
Observation: It's well known that $\frac{(2n)!}{(n!)^2}=$$2n\choose{n}$ is an integer.
Problem: Is the above true even if we decrease the numerator's factorial? I.e for some constant $k$, is there ...
- 121
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Infinitely many solution of $x^3 + y^3 + 7z^3 = 0$ on $\mathbb{Z}^3$ with gcd $=1$.
I am reading a textbook on undergraduate-level abstract algebra to have a break. I want to prove that the equation
\[
x^3 + y^3 + 7z^3 = 0 \tag{1}
\]
has infinitely many solution on $\mathbb{Z}^3$ ...
- 331
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votes
3
answers
38
views
Determining for which integers $t\in\mathbb{N}_0$, $\sqrt{t^2 - 4d}$ is an integer when $d = \pm 1$
I am currently stuck on a proof which states that if $d = \pm 1$, then the only non-negative integers $t\in\mathbb{N}_0$ for which $\sqrt{t^2 -4d}$ is an integer are $t \in \{0,2\}$. It is clear that ...
- 445
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How do I prove equality of the (sub)sets created?
We have the following two functions with n and m being positive integers:
f(n,m) = (4^m(6n-5)-1)/3
g(n,m) = (4^m(6n-1)-2)/6
Let's say f(n,m) produces dataset S1 and g(n,m) produces dataset S2. Now I ...
4
votes
2
answers
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views
Why are $p$-adic integers integers
When constructing the $p$-adic numbers, we proceed for instance as when constructing $\mathbb{R}$ for the usual distance. Then the integers are king of ``natural", we are used to them (are we can ...
- 1,175
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1
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47
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Distance from linear combinations over the integers
I'm a bit confused with the inequalities over the integers, my question is if $a\in\mathbb{R}\backslash\mathbb{Q}$ is fixed and $q\neq0$. How do I prove $|am+n+q|\geq \epsilon$ if $am+n+q\neq 0$, ...
- 1,209
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2
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48
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How to compare large integers represented as exponents
I have large integers that can only be factored into 2, 3 and 5, which are known as 5-smooth numbers. Since they are extremely large, I want to represent each such number as a triplet of exponents; ...
- 5,943
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1
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102
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Why $\bigg\lfloor \frac{n^m}{\binom{n}{m}}\bigg\rfloor=m!$ for $n>(m+1)^{m+2}$?
Let $n,m\in \mathbb{N}_0$. I need to prove that if $n>(m+1)^{m+2}$ then
$$ \Bigg\lfloor \frac{n^m}{\binom{n}{m}}\Bigg\rfloor=m!.$$
Since
$$ \frac{n^m}{\binom{n}{m}}=\frac{n^m(n-m)!}{n!}m!=m!+\bigg(\...
- 368
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why is $(-1)^{3m/2} \cdot b^{3m/2} = (-b)^{3m/2}$?
Assume m, b are positive integers.
$(-1)^{3m/2}$ can be written as $\sqrt{(-1)^{3m}}$. If m is even, then $(-1)^{3m}=1$, otherwise $(-1)^{3m}=-1$. Now $\sqrt{1}=1$ and $\sqrt{-1}=i$.
But why is then $(...
3
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1
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Proving that if $k\ge60$, then there exists an integer $t$ such that $\frac{3}{11}k \le t \le \frac{2}{7}k$
(Sorry for my bad english, first time using stack exchange) I came up with that "conjecture" solving a problem. With Geogebra I saw that is probably true. Since $\frac{2}{7}k-\frac{3}{11}k=\...
- 33
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Does for every $n>1$ exist a sum of $n$ squares of consecutive natural numbers, that equals to the sum of squares of next $n-1$ consecutive numbers?
Can we, for every natural $n>1$, find such a natural number $k$ that sum of $n$ squares of consecutive natural numbers starting with $k$, that is $k^2+(k+1)^2+...+(k-1+n)^2$, will be equal to the ...
- 546
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If $n>m>0, n, m \in \Bbb{N}$, is $n^m-m^n$ positive or negative? (Explain with cases if needed.)
If $n>m>0, n, m \in \Bbb{N}$, is $n^m-m^n$ positive or negative? (Explain with cases if needed.)
First, for finding patterns, I've put some small integers to $(n, m).$
\begin{align}
(2, 1): \; &...
- 2,392
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1
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Proving no integer solution exists that makes a polynomial a perfect square
The context for this is the following coding problem on Hackerrank. I'm trying to understand why one of their sample inputs (Sample Input 4) has no solution.
After a bit of math, it comes down to ...
- 1,596
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Why is it possible to define a partition of integers as prime, composite and {1,-1,0} although divisibility is not an equivalence relationship?
It's a known theorem that if $\mathcal{R}$ is an equivalence relation defined on a set, let's say $A$, then $\mathcal R$-equivalence-classes define a partition of $A$. It is also known that the ...
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Possible to determine the sign based on individual values?
I have a list of positive and negative integers and I wondered if there was a way to determine if the total of those values would be positive or negative, without actually summing the individual ...
0
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1
answer
60
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Queen's graph diameter and knight's graph diameter for a $n \times n \times \dots \times n$ chessboard
Let a $n \times n \times n$ chessboard be given. I have just proven (by brute force) that, the queen's graph diameter, ${d_{n}^k}(Q)$ (i.e., the number of moves needed to move a queen from any 3D cell ...
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For what integers $n$, do we have $30n+11=6x^2+5y^2$ for some integers $x,y$?
I am trying to find all integers $m$ such that $m$ is relatively prime to 30, and $m=6x^2+5y^2$ for some integers $x,y$. Note that we must have: $y$ is odd, $(y,3)=1=(x,5)$. Using these conditions, I ...
- 1,974
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Is there a formal approach to rounding of numbers?
Math formalizes arithmetic as algebraic structures and operations on those structures. Rounding numbers is a common operation. Is there a commonly-accepted formalization of that operation as well?
E.g....
- 136
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Exact fast factorial computation
For calculating a power, i.e. $a^b$, well-known fast algorithm exists, based on bitwise decomposition of $b$ and sequential squaring of $a$. Are there similar fast algorithms for calculating a ...
2
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2
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One type of integer divisor homology - what does this one measure?
The set of functions $\{ (d \mid \cdot) : d \in \Bbb{N}\}$ where $$(d\mid x) = \begin{cases} 1, \text{ if } d \mid x, \\ 0, \text{ if } d \nmid x \end{cases}, \ \forall x \in \Bbb{Z}$$
are the "...
- 19.8k
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Find all positive integers x, y
Find all positive integers x, y such that $(x^2 + y)(x + y^2) = (xy)^3$
I get the problem from a practice problem set of a math camp which was organised for our national math olympiad.
I think that ...
- 75
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The making second order polinimial equation square of an integer for a given parameter value
I am working on an important problem, I have been stuck with the following subproblem. I will appreciate it if you point out the method to solve it. Thanks.
I have the following equation ax^2+bx+c, ...
- 1
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2
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Diophantine Equations Problem
Find all positive integers $x, y, z$ such that $x^2y+y^2z+z^2x = 3xyz$.
I first tried to solve it by spilting $xyz$ to every expression and factor it. But it fails.
I notice that it is true for $\...
- 75
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A problem of integer approximation
Suppose we are given k integers n1, n2, ..., nk and a "target" integer m. We are required to get as close as possible to m using only the given k integers (each one only once) combined via ...
- 279
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Square numbers with a form of $p^{2n+1} + q^{2n+1}$, where p, q are prime numbers and n is a positive integer
Recently I came across a difficult problem:
Let $n$ be a fixed positive integer and $p, q$ be different prime numbers. Can $p^{2n+1} + q^{2n+1}$ be a square number?
When I wrote a python program, I ...
- 59
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For a given sequence consisting of a fraction of 2 functions, is there a way to know how many entries are of integer value? [duplicate]
For a given sequence $\frac{n^2+k^2}{2n+1}$ where $k$ is a given integer and $k > 0$, is there a way to calculate how many entries will be of integer value. Or, if that is not possible, know if it ...
- 37
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If $B$ is an integer matrix, does $\mathrm{det}(B^TB)$ has to be a perfect square?
Suppose $B \in \mathbb{Z}^{m \times n}$ be an integer matrix of rank $n$. I want to prove that $\mathrm{det}(B^TB)$ is a perfect square.
Of course if $m=n$, $\mathrm{det}(B^TB) = \mathrm{det}(B)^2$ ...
- 12.3k
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What is an upper bound of $(x - r)_{\pmod d} + (x + r)_{\pmod d}$ in terms of $x$ and $d$?
Let $x, r, d \in \Bbb{Z}$, $x,d \geq 1$ and $r \geq 0$. I'm wondering what an upper bound for the expression:
$$
f(x,r,d) = (x - r)_{\pmod d} + (x + r)_{\pmod d}
$$
is in terms of $x, d$ only. I ...
- 19.8k
1
vote
1
answer
256
views
Find $a_n$ such that the formula is equal to $\zeta(2)$
My question start with the observation :
$$\sqrt{e}\simeq \frac{\pi^2}{6}$$
At first glance it's not really convincing but after some work I found :
$$\sqrt{e-5\left(\frac{1}{\pi}-\frac{1}{e}\right)^{...
- 3,317
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vote
0
answers
8
views
Computing polynomial expression with a bounded space for intermediate results
Consider the problem of computing a general numeric expression over integer variables, made of sums, products, products by a constant and powers. E.g. something like $(2xy^3 + 4y)(2z + 5)^3$.
However, ...
- 163
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vote
1
answer
50
views
Is there a specific name for this counting system?
I needed a counting system that goes like in the spreadsheet below.
The columns are different representations of the same counting system.
Given a limit of X digits (5 in this example), we count by ...
- 1,054
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1
answer
40
views
There are 45 multiples of 3 between 85 and integer b. Why is the largest possible value of b not 219?
There are 45multiples of 3 between 85 and integer b. What is the largest possible value of b.
$(x-87)/3 + 1 = 45 \implies x=219$
Why is the answer 221?
user685056
1
vote
1
answer
47
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Definition of integers in higher-order logic
There's the classical of natural numbers in higher-order logic (see the introduction of this page for example).
Is there something similar for integers (elements of $\mathbb{Z}$) ? I didn't find this ...
- 11
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votes
1
answer
32
views
A formal analytical way to solve system of algebraic equations within integer domain
Let's take this famous puzzle as an example.
I can write it as a system of equations like this
$$
3(100C+10A+R)=(100R+10R+R)
$$
$$
C \in \mathbb{Z}
$$
$$
A \in \mathbb{Z}
$$
$$
R \in \mathbb{Z}
$$
$$
...
- 151
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0
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May I know why the two formula is equivalent?
This is a lemma 6 in published article called The Number of Prime Factors on Average in Certain Integer Sequences, page8. i have some trouble expounding or looking for the in between equation, before ...
- 103
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1
answer
35
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Is it possible to simplify $n \equiv - \frac d 4\bigg(\frac 3 2\bigg)^{m-2} - 2 \pmod {3^m}$ further?
Is it possible to simplify this further:
$$
n \equiv - \frac d 4\bigg(\frac 3 2\bigg)^{m-2} - 2 \pmod {3^m} \qquad \text{where } n, d \geq 0 \text{ and } m \geq 2 \text{ are integers}
$$
or to glean ...
- 1,381
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1
answer
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For which integer values of $m \geq 0$ and $n \geq 0$ is $A_m(n) = (\frac 2 3) ^ m (n + 2) - 2$ a positive integer?
For which integer values of $m \geq 0$ and $n \geq 0$ is $A_m(n) = (\frac 2 3) ^ m (n + 2) - 2$ a positive integer?
I made a table of the first few expressions for $m \in [0, 5]$
$$
\begin{align}
m &...
- 1,381
6
votes
3
answers
244
views
Prove that $2\cdot 3^x +1= p^y$ has no solution
Prove that the Diophantine equation of $3$ variables $(x,y,p)$
$$2\cdot 3^x +1= p^y$$
has no solution where $x,y\in\mathbb{N}_+$, $x\ge2, y\ge2$ and $p$ is a prime number.
I found that $y$ cannot be ...
user1124380