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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Multiple Questions about Integer Lines, Rational Lines, Unit Circle Rational Lines and Rational Angle Lines [Terms defined in Body of Question] [closed]

Defining Few Terms Integer Lines : The set of all (Real) lines I could create from $O$ to any $(x,y)$ where $x$ and $y$ are integers. Rational Lines : The set of all (Real) lines I could create from ...
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-2 votes
1 answer
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What do you call rings that have unique factorizations?

For example, integers, gaussian integers, and polynomials all have unique factorizations. What are these rings (or this property) referred to as? Or is unique factorization a ubiquitous property that ...
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7 votes
2 answers
116 views

Generating Pythagorean triples where the legs are Hypotenuses of other Pythagorean triples

I know how to generate regular Pythagorean Triples given two positive integers P and Q such that $$a=2*p*q$$ $$b=p^2-q^2$$ $$c=p^2+q^2$$ where $p>q$, but I want to find scenarios where $a$ and $b$ ...
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-1 votes
0 answers
87 views

A system of cubic diophantine equations over positive integers

I was trying to solve an exercise and it led me to this system of equations: $$a^3 + b^3 = c^3 + x^3\\ c^3 + e^3 = a^3 + y^3\\ c^3 + d^3 = b^3 + z^3\\ c^3 + d^3 + e^3 = a^3 + b^3 + t^3$$ I need to ...
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35 views

Expressing any even natural number as a sum of primorials with coefficients

I'm having a hard time trying to solve the following problem: Given any random even natural number, $x$, prove that it can or cannot be written as the product of some integer, $b$, times the primorial ...
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2 votes
0 answers
77 views

For $n\in\mathbb{Z^+}$ unique $f(n)\in\mathbb{Z^+}$ is mapped with $f(1)=1$, $f(f(n))=n$ and $f(2n)=2f(n)+1$. Find $f(2020)$.

Problem For every positive integer $n$, a unique positive integer $f(n)$ is assigned in the following manner: $f(1)=1$ and for every positive integer $n$, $f(f(n))=n$ and $f(2n)=2f(n)+1$. Find the ...
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Average decreasing non-sequential sub-list length in a random list

Let $l$ be a randomised list of integers $\in[1,r]$ of length $n$. We will construct a sequence as such : Take the first value of $l$ For each subsequent value, add it to the sequence if it is equal ...
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3 votes
2 answers
84 views

How could I prove / disprove that every non-zero integer can be written in the form $p-x^2$ where $p$ is a prime and $x$ is a positive integer?

Question: Can every non-zero integer be written in the following form? $$p-x^2$$ I was thinking about if every non-zero integer could be written in the form $p-x^2$ where $p$ is a prime and $x$ is a ...
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-3 votes
2 answers
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How do I prove that, for any positive integer $n > 2$, $n^{n/2} < n!$ [closed]

I tried using Induction, but I couldn't prove the inequality. Any proof would work. Rewriting the question for clarity, here is its statement: For any positive integer $n > 2$, prove that $n^{n/2} &...
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1 vote
1 answer
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Finding all pairs of three-digit integers that fit some conditions

I need to find all pairs of three-digit integers $(m,n)$ with $m,n \in N $that fit the following conditions: a) $m-n=889$ b) For the digit sum $Q(m)$ and $Q(n)$ let: $Q(m)-Q(n)=25$ My ideas: I tried ...
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0 votes
1 answer
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When is $n = \frac{a_{1}x^{2} + a_{2}x + a_{3}}{a_{4}x + a_{5}}$ an integer where $a_{i}$ are incredibly large integers? [closed]

I have the equation with the following form: $$n = \frac{a_{1}x^{2} + a_{2}x + a_{3}}{a_{4}x + a_{5}}$$ Where $a_{i}$ are incredibly large (e.g 1000 digits) but unrelated (not part of sequence or ...
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5 votes
1 answer
196 views

Integral Basis of $O_k$

Let $K=Q(\sqrt 6,\sqrt{11})$. Write $α ∈ O_K$ and its conjugates in terms of a $Q$-basis. And show that an integral basis of $O_K$ is given by ${1,\sqrt 6,\sqrt {11},\frac{\sqrt 6+\sqrt{66}}2 }$, from ...
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2 answers
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Prove that there doesn't exist a polynomial $P$ with integer coefficients such that $P(2)=P(-2)=2$ and $P(0)=0$.

Stumbled upon this problem in a math competition. Couldn't figure it out. Thanks
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Why do we split the Number System into Even and Odd? [closed]

I started thinking about this whilst learning that the difference between a Baryon and a Meson is their possession of an odd or even number of quarks; I couldn't think of the name for the odd-even ...
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2 votes
3 answers
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When is $K = an^2 + bn + c$ a square number?

Suppose I had the equation: $K = an^2 + bn + c$ where: $n$ is a positive unknown integer. $a,b,c$ are positive known integers. Problem: What values of $n$ make $K$ a square number? (1a) Is there any ...
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0 votes
1 answer
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Finding specific, easy, closed formula as a tutorial example

Is there a sequence in the integer numbers that has the following properties: a) the sequence has a closed formula without using fractions/devision, b) the sequence converges, c) the first m numbers ...
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3 votes
1 answer
62 views

Existence of an integer matrix with maximal subdeterminants $a_1, \ldots, a_n$

Given $n \geq 2$ and integers $a_1, \ldots, a_n$, does there exist an integer $(n-1) \times n$ matrix whose maximal subdeterminants are $a_1, \ldots, a_n$ (with fixed ordering)? Example: $n = 3$, $(...
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0 votes
1 answer
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An inequality involving box function [closed]

Let $t$ be an even integer greater than or equal to $6$ and $n$ is an odd natural number. Then is it always true that : $\lfloor\frac{5-t}{2} \rfloor+1 + \frac {t+n-3}{2} \geq 2$? I have tried for ...
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  • 863
12 votes
4 answers
351 views

Finding $(a,b)\in\mathbb{N}^2$ such that $\dfrac{a^2+b^2+1}{a+b} \in \mathbb{N}$.

A pair $(a,b)\in\mathbb{N}^2$ is called good if $a < b$ and $$\frac{a^2+b^2+1}{a+b}\in\mathbb{N}.$$ I think I've shown that there are infinitely many good pairs. However, the family of good pairs ...
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1 vote
0 answers
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Vector of 16bit Integers

I would like to define a vector $\mathbf{s} = [s_1, \ldots, s_N]^{\operatorname{T}}$ with length $N$ where each element is a 16bit two's complement signed integer (in C one would write ...
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1 answer
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Divisibility proof of "If $a\mid b, \text{and}, a\mid c, \text{then}, a \mid (mb + nc)$

I would just like to make sure my proof of the statement: If $a,b,c,m,n \in \mathbb{Z}, a\mid b, \text{and}, a\mid c, \text{then}, a \mid (mb + nc)$ Makes sense and is correct. If $a \mid b$ and $a ...
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0 votes
1 answer
39 views

Why is this recursively-defined permutation $\sigma:\mathbb{N}\to\mathbb{N}$ symmetrical in the sense that $\sigma(\sigma(n))=n\ $?

Define a permutation $\sigma:\mathbb{N}\to\mathbb{N}$ recursively by the following rule: $\sigma(1)=1$ and $\sigma(n)$ is the least $x\ $ in $\ \mathbb{N}\setminus\{\sigma(k): k\leq n-1\}$ such that $...
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0 votes
0 answers
42 views

Hints and tips to prove this inequality.

Let's say that $n = \lfloor{2^{\frac{k}{2}}}\rfloor$ for some integer $k \geq 3$. I need to prove this inequality: $${n \choose k}\cdot2^{1-{k \choose 2}} < 1$$ I tried to rewrite those terms or ...
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How to solve a quadratic Diophantine equation without trial and error by just inputting integer values for one of the variables?

I’m looking for a way to find only integer solution pairs to a dual-variable quadratic equation without trial and error. For example: $$(a+3\sqrt 5)^2+a-b\sqrt 5=51$$ Valid solution pairs are any ...
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2 votes
0 answers
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Representations of negative numbers as a positive offset from a multiple of 10

In decimal, adding 10 to a positive number leaves the units digit unchanged, as does subtracting 10 from a negative number. This is also true when adding or subtracting integer multiples of 10, so ...
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3 votes
2 answers
105 views

Why $ \sum _{k=0} ^{n-1} \binom{n-1}{k} x^{k+1} = \sum _{k=1} ^{n} \binom{n-1}{k-1} x^k $?

I'm struggling to understand how to get from this: $$ \sum _{k=0} ^{n-1} \binom{n-1}{k} x^{k+1} $$ to this: $$ \sum _{k=1} ^{n} \binom{n-1}{k-1} x^k $$ I always have a problem understand the ...
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8 votes
2 answers
124 views

How many associative binary operations on the integers does $+$ distribute over?

I am interested in binary operations $\mid: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ which satisfy: Associativity: $a \mid (b \mid c) = (a \mid b) \mid c$ $+$ distributes over $\mid$: $(a \mid b) ...
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  • 44k
0 votes
0 answers
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Terminology for a "basis of an integer set"

I am trying to find a name for the following concept: Let $A = \{a_1, a_2, \dots a_n\}$ be a set of positive integers. Then a set of positive integers $B = \{b_1, b_2, \dots b_k\}$ is something of A ...
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3 votes
2 answers
99 views

Prove that $\dfrac{nm^2-n+1}{2mn-2n+1} \notin\mathbb{Z}$ when $m \geq 2$

Given: $n$ and $m$ are positive integers with $m≥2$, show that: $$\frac {nm²-n+1}{2mn - 2n+1}\notin \mathbb Z$$ My attempt: $n, m \in \mathbb{Z}$ $nm^2 \in \mathbb{Z}$ $nm^2 - n \in \mathbb{Z}$ $2mn - ...
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1 vote
1 answer
42 views

Tangential Equidiagonal (Irregular) Quadrilateral with integer coordinates

Playing with mathematics this week-end, I ended up with this question I couldn't solve, so I'm asking here. I try to find a Quadrilateral ABCD with integers (cartesian) coordinates (points are on a ...
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0 votes
0 answers
8 views

When do differences between pairs of integers in a set occur exactly once at even intervals

Consider a set $S$ containing $n$ integers, and construct a multiset $D = \{x - y: x > y, \,\,\,x,y\in S\}$ that contains every strictly positive difference that can be computed by subtracting two ...
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1 vote
1 answer
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Are there more integer solutions to this system of equations?

Let $n>0$ and $b_1,b_2, \dots, b_n,b_{n+1},b_{n+2} >0$ as well as $c_1, \dots, c_n >0$ be positive integers. We ask that $b_i < c_j$ for $i \in \{1,\dots,n+1\}$ and $j \in \{1, \dots, n\}$....
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0 votes
1 answer
23 views

On pairs of integers satisfying an inequality

Let us consider the following set $A:=\{(r,t)| r \in \mathbb N \cup \{0\}, t \in \mathbb Z, t \leq r-5\}$ My question is the follow : Does there exists any pair $(r,t)$ belonging to $A$ such that it ...
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4 votes
2 answers
103 views

Any algebraic number with modulus $1$ is root of a polynomial with positive coefficients

Given a complex number on the unit circle $e^{i\theta}\neq 1$ that is the root of some polynomial in $\mathbb Z[x]$, can we always construct a polynomial $p(x)$ with positive integer coefficients such ...
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  • 136
0 votes
2 answers
46 views

Integral rational over $ \mathbb Z$ is an integer

I want to prove that if a rational $y\in \mathbb Q$ is integral over $\mathbb Z$, then it is an integer. We say that $y$ is integral over $\mathbb Z$ if there exists a monic polynomial $F\in \mathbb Z[...
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4 votes
3 answers
273 views

Finding Integer Approximations

The Saros is the time period for the draconic month ($T_d$ = 27.212220815 days), synodic month ($T_s$ = 29.530588861 days) and anomalistic month ($T_a$ = 27.554549886 days) to approximately match. ...
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  • 541
1 vote
0 answers
29 views

Lower bound on number of integers in interval which are coprime to every other integer on interval

Given a set $X = \{x | L \leq x \leq R\ \land\ x \in \mathbb{N}\}$ (and $L\leq R$) I need to find an approximate lower bound on number of integers $a \in X$ such that they are relatively coprime to ...
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2 votes
1 answer
56 views

Axiomatization of $(\mathbb Z, <)$

I'm interested in the axiomatization of the total order $(\mathbb Z, <)$. My idea is to have first the axioms for a total order: $\exists x : x = x$ $\forall x : \lnot(x < x)$ $\forall x : \...
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1 vote
2 answers
73 views

Perturbing integer roots

The polynomial $$(x-a_1)(x-a_2)\cdots(x-a_n) - x^{n-1} = 0$$ for arbitrary integers $a_i$ has come up in a project. Is the root with the greatest modulus always real? Evidence from several choices ...
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1 vote
1 answer
82 views

Find all solutions in set of integers for $a^2 + 5b^2 = 3c^2$

I haven't learnt Diophantine equations to solve this equation (just know Modular Arithmetic). So given the equation we are to solve for all solutions in $\Bbb Z$. So what I've done so far: Case 1: $\...
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0 votes
0 answers
33 views

Simple Proof over integers and rational numbers.

I'm sure this is a relatively easy thing to do, I'm just not sure how to begin. Here I wish to show that if $k \in \mathbb{Z}$ that $$ \frac{3}{2} \pm k \not \in \mathbb{Z} $$ Or, to extend, if $p,q \...
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  • 23
2 votes
1 answer
46 views

Equation with polynomial with integer coeficients.

Let $p>3$ be a prime number. Prove that there doesn't exist a pair of polynomials $(f,g)\in{\mathbb{Z}[X]\times\mathbb{Z}[X]}$ such that: $X^{2p}+pX^{p+1}-1=[(X+1)^p+p\cdot f(X)]\cdot[(X-1)^p+p\...
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  • 758
1 vote
0 answers
48 views

Proving that every ring $R$ admits a unique homomorphism $\mathbb{Z} \to R$ [duplicate]

I am trying to prove that there is a unique homomorphism between every ring, R and the integers, $\mathbb{Z}$. I suggested that such a homomorphism $\phi : \mathbb{Z} \to R$ could be defined by for $ ...
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0 votes
0 answers
44 views

Finding restriction of the ultraproduct that behaves like $\mathbb{Z}$

Let $\mathcal{A}= \prod_{n \in \mathbb{N}} \mathcal{A}_n /\mathcal{U}$, where $\mathcal{A}_n=(\{0, 1, \dots, n\},<)$ and $\mathcal{U}$ is a non-principal ultrafilter of $\mathbb{N}$. Can we find ...
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1 vote
0 answers
53 views

The sets $\mathcal{F}_d(m,n,k) = \Big\{ x \in (m, n) : x^2 = k^2 \pmod d\Big\}$ seem to have relationships with each other. What is their structure?

For $m,n,k,d\in \Bbb{Z}, m\leq n$, define for interval of integers $(m,n)$: $$ \mathcal{F}_d(m,n,k) := \Big\{ x \in (m,n): x^2 = k^2 \pmod d\Big\} $$ Then $\mathcal{F}_d(m,n, i)\cdot\mathcal{F}_d(m', ...
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4 votes
1 answer
169 views

Prove sequence has optimal length

Let there be a sequence of integers such that each term of the sequence is the sum of two previous terms, not necessarily distinct. That is, for sequence $a_i$ with length $\ell$, we require $a_1=1$ ...
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1 vote
0 answers
121 views

Multiplication on the Integers Proof

From the construction of the integers from the naturals, I am trying to prove that multiplication on the integers is well defined. Is this valid? I'm using the following definitions: On the set $\...
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0 votes
0 answers
13 views

Smith normal form of a real symmetric matrix

I have a real integer symmetric matrix $A$ for which I know has eigenvalue decomposition $$A=QDQ^T$$ I know that $D$ is a vector of integers, but $Q$ is an orthogonal matrix consisting of real, ...
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  • 411
0 votes
2 answers
51 views

Subgroups of the additive groups of integrers mod n

I'm currently working in group theory , following Hugerford's algebra chapter 1, and I was seeing the subgroups of the integers mod n under addition in an example in the section 1.2 and a question ...
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0 votes
2 answers
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Understanding the Number Line through Convexity

I came across the following picture: According to this picture, it would appear that Natural Numbers, Integers, Rational Numbers and Real Algebraic Numbers are all Non-Convex sets - but Real Numbers ...
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