# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
1 vote
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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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### 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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### 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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### 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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1 vote
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### 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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### 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 ...
1 vote
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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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### 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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### 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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1 vote
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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, ...