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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11
votes
0answers
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 ...
10
votes
0answers
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 ...
9
votes
0answers
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 ...
8
votes
0answers
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 ...
7
votes
0answers
78 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
0answers
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 ...
6
votes
0answers
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. ...
6
votes
0answers
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 ...
6
votes
2answers
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: ...
6
votes
1answer
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 ...
5
votes
0answers
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 $\forall$ $1\leq i,j,k\leq n$ ...
5
votes
0answers
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 ...
5
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
86 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
votes
0answers
60 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
89 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
148 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
163 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
votes
0answers
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?
3
votes
0answers
63 views

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

I have the following problem: Prove that for a free choosable value of $x$, where $x\in\mathbb{N}$ and $x>7$, that the following equation has no positive integer solutions for $(k,n)$, where $k\in\...
3
votes
0answers
36 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
124 views

Consider x and y positive integers. Which is the smallest positive integer value that $|11x^5−7y^3|$ could assume?

Consider x and y positive integers. Which is the smallest positive integer value that $|11x^5−7y^3|$ could assume? I tried to separate in two cases. First, with $11x^5\geq 7y^3$. Clearly, the ...
3
votes
0answers
52 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 ...
3
votes
0answers
50 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
132 views

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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
70 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
1answer
68 views

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 ...
3
votes
0answers
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,...
3
votes
0answers
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?
3
votes
0answers
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 ...
3
votes
0answers
82 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
142 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
339 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
112 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
46 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
1answer
176 views

Metrics on integers

I am looking for a list of distances that are defined on the set of the positive integers. I am mostly interested in metrics that make the set complete, but I also consider other metrics. Any ...
3
votes
0answers
85 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
415 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
161 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 ...
2
votes
0answers
42 views

If I keep on multiplying a number by itself, will I eventually get really close to a whole number?

Main Question If we look at the sequence $$x^n \bmod 1$$ for increasing values of $n$ where x is a real number, are we guaranteed that we can find some value in the sequence arbitrarily close to 0? ...
2
votes
0answers
57 views

Find integer solution $|x^2-y|=8|y^2-1|$

Find integer solution $|x^2-y|=8|y^2-1| (1) $ I try to: If $y^2-1=0$ we have $y\in\{1,-1\}$ With $y=1\Longrightarrow x=\pm 1$ $y=-1\Longrightarrow x^2+1=0$ not satisfied. In other case: $\left| \...
2
votes
1answer
31 views

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 ...
2
votes
0answers
42 views

Proof that [(1 + √3) ²ⁿ⁺¹] is divisible by 2ⁿ⁺¹ ( [x] denotes the greatest integer function of x) for n >= 0

I came up with a proof but I am not sure if that is correct. I am not sure whether this is rigorous proof, but I think I have a proof for the fact that $[(1 + \sqrt{3})^{2n+1}] = k2^{n+1}$ for $k \in ...
2
votes
0answers
43 views

Divisors of a multiple of a prime (or any integer)

I have been studying group theory for some time now, and I have noted that quite a few theorems/proofs considering finite groups rely on results from number theory, a branch of mathematics of which I ...
2
votes
1answer
62 views

Proof explaination - $\sum_{i=1}^{n} \frac{1}{i}$ is not an integer for $n>1$

I was reading a proof to the following fact: for $n>1$, $\sum_{i=1}^{n} \frac{1}{i} \notin \mathbb{Z}$. The proof is as follows: Denote for prime $p$ by $v_p(a)$ the p-adic valuation of $a$. Write ...

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