# 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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### 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 ...
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### 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 ...
512 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 ...
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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 ...
363 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 ...
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### 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 ...
82 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 ...
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### 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. ...
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 ...
178 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: ...
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### 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 ...
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### 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 ...
211 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 $k^n + (k+1)^n + (k+2)^n = (k+3)^n + (k+4)^n$ ...
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?
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### How to find numbers like 174

We know Ramanujan number: $1729 = 1^3+12^3 = 9^3+10^3$ The smallest number expressible as the sum of cubes of two positive integers in two different ways. We also know how to find other Ramanujan ...
119 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 ...
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### 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\{...
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 $n>4$...
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### 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 $ab=0,$ ...
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### 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" ...
53 views