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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11
votes
0answers
207 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 ...
10
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0answers
346 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 ...
9
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0answers
514 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 ...
8
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0answers
380 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 ...
7
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0answers
105 views

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 ...
6
votes
1answer
71 views

Two bijections between set of integers

I have the following interesting question: Let $b$ be a $\mathbb{Z} \rightarrow \mathbb{Z}$ bijection, where $\mathbb{Z}$ denotes the set of integers. Is it possible that there exist a bijection ...
6
votes
0answers
85 views

How many unique values of $\cos(\frac{a\pi}{N})\cos(\frac{b\pi}{N})$ are there for the positive integers $a,b < N$

The Question How many unique values of $\cos(\frac{a\pi}{N})\cos(\frac{b\pi}{N})$ are there for the positive integers $a,b < N$ for a given $N$? I would like a function $f(N)$ which gives that ...
6
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0answers
237 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. ...
6
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0answers
157 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 ...
6
votes
2answers
179 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: ...
6
votes
1answer
149 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 ...
5
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0answers
35 views

Properties of integers squareness function

Consider the map $$ \begin{array}{rccc} s: &\mathbb{N}_{\geq1} & \longrightarrow & \left(0,1\right]\cap\mathbb{Q}\\ & n & \longmapsto & \frac{\max\{d|n:\ d\ \leq\ \sqrt{n}\}}...
5
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0answers
65 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 $\forall$ $1\leq i,j,k\leq n$ ...
5
votes
0answers
143 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 ...
5
votes
0answers
154 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 ...
4
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0answers
77 views

Isomorphism between different definitions of Integers

Sorry for the wall-of-text in advance, hope some of you make it till the end ;) When constructing the integers from the natural numbers, I've been using the following definitions. Let $\sim$ be an ...
4
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0answers
110 views

Sum of rising numbers, strange pattern

I recently tackled the following problem, in a coding competition: Let a rising number be a positive number whose decimal representation has no digit smaller than a previous digit. For example, 445678 ...
4
votes
1answer
111 views

binomial identity seemingly illogical and impossible. Is there any way it could be true?

There is binomial expression(s) written as $$\sum_{n\geqslant0}\frac{(-3n+2k-3)n!^2}{2(2n+1)(k-1)!^2(n-k+1)!^2 \binom{2n}{n}}=\begin{cases} 0 & \text{if $k=0$,} \\ -1 & \text{if $k\geqslant1$,...
4
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0answers
60 views

Is it possible to generate a composite number with no information about its factors?

Are there functions or algorithms which can generate integers which are necessarily composite, yet not yield any information about what factors it has? For example, $f(x):=x^2-1$ is not what I'm ...
4
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0answers
215 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 ...
4
votes
0answers
89 views

For which even integers $k$ has $\varphi(n+1)-\varphi(n)=k$ a solution?

For which even integers $k$ does the equation $$\varphi(n+1)-\varphi(n)=k$$ have a solution ? $\varphi(n)$ denotes the totient function and $n$ is a positive integer. For the following $|k|\le 1\ ...
4
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0answers
69 views

Positive integers satisfying: for all odd prime powers $p^k < n$, $n - p^k$ is prime

Given positive integer $k$, define the subset $S(k)$ of positive integers $n$ where for every odd prime power $p^k < n$, $n - p^k$ is prime. In other words, $$S(k) = \{n \in \mathbb{N} \mid \forall ...
4
votes
0answers
93 views

When does an equation of the form ${1 \over p}{(2^{p-1}-1)} = 2pxy+x+y$ have no integer solutions?

Specific equations of the form below (for different given values of p, a prime number) will either have positive integer solutions for $x$ & $y$, or will not have any integer solutions. $${2^{p-1}...
4
votes
0answers
157 views

Almost-Linear Sequence of Positive Integers Excluding a Near-Quadratic Integer Sequence

I remember that I have seen a similar problem to this one here: The set of natural numbers that don't belong to a set (which is a duplicate of $m$ doesn't come in the sequence $a_n=[n+\sqrt{n}+...
4
votes
0answers
169 views

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 $k^n + (k+1)^n + (k+2)^n = (k+3)^n + (k+4)^n$ ...
4
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0answers
120 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?
3
votes
0answers
120 views

Diophantine with factorials

This is a problem I encountered on a competition Discord server, apparently, there is an elementary solution, but I'd honestly be fine with any solution. Wolfram Alpha solves the problem, here's the ...
3
votes
0answers
46 views

Given a finite set of complex numbers, is there a name for the set generated by taking all non-negative integer linear combinations of the elements?

Many apologies if this is elementary, but consider the following. Let $D=\{z_1,\ldots,z_d\}$ be some set of $d$ distinct complex numbers. Is there a name for the following set: \begin{align*} S=\left\{...
3
votes
1answer
82 views

Solve a Diophantine equation for a specific value of $x$

I have the following problem: Prove that there exists some $x\in\mathbb{N}$ with $x>7$, such that the following equation has no solution $(k,n)$, where $k,n\in\mathbb{N}$ with $k>3$ and $n>4$...
3
votes
1answer
973 views

Is my proof for if ab = 0 then either a=0 or b=0, when we are using integers, correct?

I'm very new to this and I'm not sure if this is a proper proof. It's within the integer set so I can't use the inverse multiplication proof. Want to show that if $a,b \in \mathbb{Z}$ and $ab=0,$ ...
3
votes
0answers
39 views

By which scheme should I add the elements in series $(\sum n^{-2})^2$ and $\sum n^{-4}$ to show their rational equivalence?

We know that $\sum n^{-2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots=\frac{\pi^2}{6}$ and $\sum n^{-4}=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\dots=\frac{\pi^4}{90}$ from "high mathematics" ...
3
votes
0answers
53 views

The Airy functions and miscellaneous sequences of odd integers without repeated prime factors

I'm curious about the following miscellaneous conjectures, for which I hope that one can to get a counterexample. I add the encyclopedia Wikipedia's article for the Airy functions $\operatorname{Ai}(...
3
votes
0answers
284 views

Why is 4 rare in the Van Eck sequence?

The Van Eck sequence is defined here: https://oeis.org/A181391 By far the most frequent integer at the beginning of the sequence (up to first 10 million terms) is zero. Intuitively, one would ...
3
votes
0answers
972 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 ...
3
votes
0answers
17 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 ...
3
votes
0answers
98 views

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, ...
3
votes
0answers
113 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,...
3
votes
0answers
148 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?
3
votes
0answers
83 views

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 ...
3
votes
0answers
131 views

Largest Nontrivial Integer Definition

The largest Fermat number with a known factor is $F_{3329780}$ with prime factor $193\times2^{3329782} + 1$ At googology are many examples of huge integers. In general, for any integer $a$, it's ...
3
votes
0answers
171 views

Making a conclusion about a $\liminf $ of an increasing sequence of integers

I have a strictly increasing sequence of positive integers $\left\{ v_{n}\right\} _{n\geq1}$ for which $\sum_{n=1}^{\infty}\frac{1}{v_{n}}=\infty$ and $\liminf_{n\rightarrow\infty}\frac{v_{n}}{n}=\...
3
votes
1answer
59 views

How do I find a list (size n) of integers where the root-mean-square of the list is an integer?

I already found this one, but it discusses mostly brute force. Brute force is possible of course, but are there any other ways? Is there a way to find all lists? Is there a way to find out how many ...
3
votes
0answers
115 views

Prove that $x^2-x-1 = F_{12m + 7}$ has no solutions.

Let $\{F_n\} -$ Fibonacci sequence: $F_1=F_2=1, F_{n+1}=F_n+F_{n-1}, n\ge2$. Prove that $$x^2-x-1 = F_{12m + 7}$$ has no solutions. $x\in \mathbb N$ My work so far: $$F_{12m+7}\equiv5(\bmod8),$...
3
votes
0answers
390 views

Non-zero elements of skew-symmetric matrices

Let $A \in \mathfrak{so}(n,\mathbb{Z})$ be an integer-valued skew-symmetric matrix. Is there an equivalent matrix $A' \in \mathfrak{so}(n,\mathbb{Z})$, s.t. the number $m$ of non-zero entries is ...
3
votes
0answers
117 views

Quadratic Residues For Odd Modulo

Say I have the formula $$k^2 \equiv b^2 - 4ac \pmod n$$ where are variables are integers and $n$ is odd. So then my question is, if $b^2-4ac$ and $n$ are constant, when is there never a $k$ that will ...
3
votes
0answers
47 views

Is there a name for the least exponent $e$ such that a power of an integer is divisible by another?

Say the primes dividing $m$ also divide $n$. Is there a name for the least exponent $e$ such that $m | n^e$? I can write that explicitly using the prime factorizations of $m$ and $n$, but am ...
3
votes
0answers
87 views

Find naturals that are sum of numbers with the same digits in inverse order

In a test I've found the following exercise: We say $n \in \mathbb{N}$ is reflexive if is the sum of two naturals $x$ and $y$ such that $y$ has the same digits of $x$ witten in the inverse order (...
3
votes
1answer
522 views

Universal property of natural number semi-ring

I asked a question similar to the one I am about to ask, and I think I got a satisfactory answer. However, this time I have some more specific question. Let a semiring $(R,+,\times)$ be an algebraic ...
3
votes
1answer
170 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
3
votes
2answers
111 views

Smallest $k$ Such that $13 + 4 \cdot k \cdot p^2$ is a Perfect Odd Square

Given a prime number $p$, I am looking to find the smallest positive integer $k$ such that the following equation $$13 + 4 \cdot k \cdot p^2$$ produces a perfect odd square. All variables are integers....

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