# Questions tagged [integers]

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

1,552 questions
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### Integer $p$-norm minimzation

Suppose that, $$x^*=\underset{x\in\Bbb{Z}^n}{\operatorname{argmin}} \left \{ \Vert{x}\Vert_p \right\},$$ where, $$\Vert{x}\Vert_p=\left(\sum_{i=1}^{n} \vert{x_i}\vert^p\right)^{1/p}.$$ Can we say ...
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### Find all the values of the paramater 'a', for which the domain of the function contains only one integer.

Recently I have been studying high school mathematics in Russia. Here they have classification tests that allow you to get into the top universities. In these tests, there are some interesting ...
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### Solve $x^y + y^x = 499$ for positive integer solution

I'm asked to solve this equation $$x^y+y^x=499$$ only positive integer solutions are permitted. First I found the apparent solutions $x=498, y=1$ and $x=1, y=498$. I want to look for a way to solve ...
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### Does this equation yield only primes?

Interested in solving this equation for $x$: $\exp\Big(\frac{n}{\ln(\pi(x))}\Big)=\pi(x)$ for $n=1,2,3,...$ For $n=1$ up to $n=9,$ I got $x=5,11,13,19,29,37,47,59,73.$ $\pi(x)$ is the prime ...
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### I have the idea of using the transitive property and/or the the integer combination property. I am stuck tho.

\begin{equation} a, b, \text { and } c \text { are integers. Prove that if } a |(b-1) \text { and } 5 a |(c+2), \text { then } a |(2 b+c) \end{equation}
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### Elegant Proof that $m | xn \implies \frac{m}{(m,n)} | x$ [duplicate]

I have a proof that shows $m | xn \implies \frac{m}{(m,n)} | x$ which leans heavily on prime factorizations. Is there a more straightforward proof? Edit With this question, I was looking for a proof ...
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### Matrix of integer powers

Is there a name for the square matrix ($j=0...n$, $k=0...n$) $M_{jk} = j^k$ (with special case $M_{00}:=1$) and is there a closed general formula for its inverse? I have stumbled upon this while ...
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### 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 $\forall$ $1\leq i,j,k\leq n$ ...
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### Convert a real range to an integer range.

Let's say that we have a set of integers in the range $[1, 4]$. Now, I have a function that will calculate a distance between two vectors, and this function returns a real number in the range $[0, 1]$....
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### How to algebraically mirror a finite subset of integers? [closed]

Let's take the set $S=\{0,1,...8,,9\}$ as an example. By mirroring I mean creating a function $f$ such that $f(x) = x$ is $x \in \{0, 4\}$ and $f(x) = 4 -(x \equiv 5)$ otherwise. The above however ...
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### With this expression, which values of n gives integer results?

I need to know when would this equation give integer values, I think there might be an easy method I am not aware of, so I am asking here to know if such method/technique is known for finding a ...
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### Generators for $\mathbb{Z_n}$

I would like to show that $K$ is a generator for $\mathbb{Z}_n$ $\iff$ $\gcd(K,n)=1$ and $1 \leq K <n$. My Attempt: Assume $\gcd(K,n)=1$ and $1 \leq K <n$. That means $K \in \mathbb{Z}_n$ and ...
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### 2 distinct integers between 5 and 17 inclusive are chosen. What is the probability that their product is odd?

"Suppose two distinct integers are chosen from between 5 and 17 inclusive. What is the probability that their product is odd?" I can't figure out the probability, although I do know that both ...
3answers
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### Inverse of an Integer Matrix

I found a problem on the Open Problem Garden which asks about the conditions on a rectangular, full-rank, integer matrix such that its right inverse (given by: $A^T (AA^T)^{-1}$ ) is also an integer ...
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### Subgroups of the integers.

Theorem: The subgroups of $\mathbb{Z}$ are $n \mathbb{Z}$ for $n=0,1,2....$. My Proof: Let $H$ be an arbitrary subgroup of $\mathbb{Z}$. Let $x \in H$. If $x<0$ then since $H$ is closed under ...
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### Replacing a natural number containing a certain digit with the sum of two without that digit

A question in Google Code Jam 2019 qualification round wanted a positive integer n which contains at least one digit 4 to be represented as a sum of two positive integers a and b, neither containing 4....
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### Near integers in powers of binomials with radicals

This question comes out of a mathematics calendar problem that asked for the tenths digit of the expression $(17 + \sqrt{280})^{17}$. The calendar implied the digit should be 9, but after playing ...
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### If f = (x/y) + (y/x) +1/(xy) is an integer. Prove that f must be of the form 3x [closed]

I have tried using Induction method but I am unable to resolve it to a single variable. Also, $x$ & $y$ are positive integers. $f\:=\:\frac{x}{y}+\frac{y}{x}+\frac{1}{xy}$ Edit : This one is ...
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### Division with replacement of floating-point arithmetic to integer arithmetic

The issues: Not all the hardware has an FPU => not possible to use floating-point arithmetic. Not all the hardware has an uint64_t => not possible to use ...
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