# Questions tagged [integers]

For questions about the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$.

363 questions with no upvoted or accepted answers
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192 views

### When does $2^n$ start with n?

A trick my dad taught me for easily referencing powers of 2 is that $2^6=64$ and $2^{10}=1024$, because $64$ starts with $6$ and $1024$ starts with $10$, and so it's faster than manually doubling ...
327 views

### Let $|A|=|B|=|C|=n$ be three finite sets of integers. Find $\min |\{ab+c \mid a \in A, b \in B, c \in C\}|$.

For a triple of sets of integers $A,B,C$ with $|A|=|B|=|C|=n$, we can compute the set $S_{A,B,C} = \{ab+c \mid a \in A, b \in B, c \in C\}$. I am interested in the minimum sized $S_{A,B,C}$ when ...
486 views

### An interesting problem which only needs elementary number theory

A problem about elementary number theory While writing my paper, I came across the following problem: (all the discussion assume that $q$ is prime and $\alpha$ is a positive integer. ) We first ...
232 views

### Can $\sqrt[n]{\sqrt{a}+\sqrt{b}}+\sqrt[n]{\sqrt{a}-\sqrt{b}}$ be an integer?

The number $\sqrt{a}+\sqrt{b}$ cannot be an integer if $a,b$ are integers such that $\sqrt{b}$ is not an integer. (In fact, this is true for any number of square roots, and I believe even for cube ...
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### If $U_a = \{a n^2 - 1 : n \in \Bbb{Z}\}$ is a topological basis on $\Bbb{Z}$ then $U_a$ is clopen?

Let $P =$ the prime numbers in $\Bbb{N}$. Define $P^i = \{ \pm q_1 \cdots q_i : q_j \in P\}$. These sets $\{ P^j\}_{j\geq 0}$ with $P^0 := \{\pm 1\}, P^{-1} := \{0\}$ form a basis for a topological ...
231 views

### Closed form bijection between integers and pairs thereof

I know that it's simple enough to map the integers, $\mathbb{Z}$, to pairs of integers, $\mathbb{Z}^2$, in a bijective way (i.e. a one-to-one mapping). You can wrap the integers around the origin of ...
156 views

### Multiplicatively closed subsets of Z

Recently my kids and I ran across the Brahmagupta-Fibonacci identity and noticed that the set consisting of integers that are expressible as the sum of two squares is closed under multiplication. ...
149 views

### Integer divisibility

Given a (not strictly) decreasing sequence of natural positive numbers $a_1, a_2, \dots, a_n$ prove that $$\prod_{i<j} j-i \quad\big|\quad \prod_{i<j} a_i - a_j - i +j$$ I already know a ...
158 views

### Solving a Diophantine equation3

The Diophantine equation that I have to solve is: $$343x^2-27y^2=1$$ This question has already been posted by other user but it has not received an answer. I proved to solve it. This is my attempt: ...
143 views

### Prove that if $2^x,3^x, 5^x, 7^x, 11^x …$ are all integers then $x$ is an integer as well

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ...$ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
61 views

### Find $n$ integers from $3n$ ones

$n$ is a positive integer. Is the following statement true? For any $3n$ integers, saying $\{b_1,..,b_{3n}\}$. There exists $n$ of them, saying $\{a_1,..,a_n\}$,so that $\forall$ $1\leq i,j,k\leq n$ ...
91 views

### Generate $1$, …,$n$ as pairwise differences of a set of numbers

Given $n \in \mathbb N$, I'm interested in sets $S \subset \mathbb Z$ such that the numbers $1$, $2$, ..., $n$ can be obtained as differences between pairs of elements in $S$. I'd like to find the ...
147 views

### Is $63\times63$ the largest matrix with no rectangles that have an even number of each $0-9$ digit?

I'm working on an algorithm that finds the largest rectangle with an even number of all 10 digits within a square matrix, e.g. for the 4×4 square on the left that would be this 3×2 ...
120 views

### When should Pollard Rho be used as opposed to trial division?

How does one know which factorization algorithm to use on a given integer? More specifically, when should Pollard Rho Brent be used instead of simple trial division? Given the random nature of Pollard ...
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### Sum of powers of consecutive integers

The equation $k^2 + (k+1)^2 + (k+2)^2 = (k+3)^2 + (k+4)^2$ has unique positive integer solution $k = 10$. For which integers $n > 2$, if any, does $k^n + (k+1)^n + (k+2)^n = (k+3)^n + (k+4)^n$ ...
116 views

### Is $\frac{\pi}{e}$ an algebraic integer?

From what I know, it is still an open question whether or not $\frac{\pi}{e}$ is irrational, but is there a proof that $\frac{\pi}{e}$ is not an algebraic integer?
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### What is the correct name for a non-whole real number?

I apologise for the simple nature of this question but I can't find the answer. I know that a whole number is an integer. I also know that a number that can be expressed as the quotient of two ...
266 views

### Simple Proof of Binomial Theorem for Negative Integer Powers

There's a vast amount of clutter on the internet about this which I've been trawling through but it does not answer exactly what I'm asking which, because superficially similar questions have been on ...
16 views

### Which polynomial forms generate non-trivial semigroups of integers under multiplication?

It's a trivial result that if $a,b$ are integers then there is an integer $c$ such that $c^2 = (a^2)(b^2)$ A slightly deeper result is that if $a,b$ are integers such that $a = u_a^2 + v_a^2$ (for ...
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### Quotient cancellation for invertible ideals of orders in quadratic fields

Let $K/\mathbb{Q}$ be an imaginary quadratic field, $m\ge 1$ be a positive integer and let $\mathcal{O}=\mathbb{Z}+m\mathcal{O}_K\subset \mathcal{O}_K$ be the unique order of index (equivalently, ...
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### Representation of an integer / Euclidean algorithm

Let $r \in \mathbb{N}$ be a natural number. Let $$L \geq 2(r-1)²$$ A paper (on quantum information theory, I'm not an expert in number theory or so...) I'm recently reading now says "One can easily ...
108 views

### If $\displaystyle\frac{x^2}{4}$ is an integer greater than 50, then what is the smallest possible value of $x^2$

If $\displaystyle\frac{x^2}{4}$ is an integer greater than 50, then what is the smallest possible value of $x^2$ ? IMO the answer should be $204$, the next small integer after $50$ is $51$, therefore,...
130 views

### Permutations of the first $n$ positive integers

What is the largest n such that there exists a permutation of the first n positive integers in which no two consecutive terms share any digit?
156 views

### An integer sequence related to Penrose tessellation

Consider covering the plane by means of the classical Penrose tiles (i.e. the "fat" and "thin" rhombi) in a spiraling fashion, adding step by step a new tile around a given one, as introduced in this ...
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### Find all functions satisfying certain requirements

The requirements are: $f(x, y) = f(y, x)$ $f(x, x) = x$ $f(x, y) = f(x, x + y)$ f: $\mathbb{N}^2 \rightarrow \mathbb{N}$ I think $\gcd(x, y)$ works, but haven't found any other solutions nor have I ...
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### Irreducibility of polynomials degree $3$
Let $f(x) = 2x^3+ax^2+bx+c$ where $a,b,c \in \Bbb Z$. Prove that $f$ is irreducible in $\Bbb Q[x]$ if and only if $f(d/2)$ does not equal $0$ for all $d \in \Bbb Z$. I'm not really sure how to start ...