# 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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### Probability that two random numbers are coprime is $\frac{6}{\pi^2}$

This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is ...
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### What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
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### A polynomial with integer coefficients that attains the value $5$ at four distinct points

There is a polynomial $f$ of integer coefficients such that $\deg(f) \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no integer $k$ ...
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### Reducibility of $x^3+nx+1$ over $\Bbb Z$

For what values of $n$, where $n$ is an integer, the polynomial $x^3+nx+1$ is reducible over $\Bbb Z$. My attempt: When $n= 0,-2$, the given polynomial is reducible over $\Bbb Z$ as $x=-1$ and $x=1$ ...
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### What three odd integers have a sum of 30? [duplicate]

I've been asked the following question: What three odd integers from the set {1,3,5,7,9,11,13,15} that when summed together equals to 30? Note that any integer can be used more than once. Is there ...
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### Is $\mathbb{Z}$ the only totally-ordered PID that is "special"?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ ...
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### Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$

I am having difficulty proving the following inequality: $$\sqrt[n]{n} < 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n.$$ I am trying to use mathematical induction but I ...
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### Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$ [duplicate]

I started like this : $a^2+c^2=b^2(a^2-1)\\c^2 +1=(a^2-1)(b^2-1)$ but it's leads to nowhere. can you help please ?
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### Can we prove the existence of a gcd in $\mathbb Z$ without using division with remainder?

For $a,b\in\mathbb Z$ not both $0$, we say $d$ is a gcd of $a$ and $b$ if $d$ is a common divisor of $a$ and $b$ and if every common divisor of $a$ and $b$ divides $d$. With this definition, can we ...
$a,b,c,x,y,z\in \mathbb Z$ ,they are all positive. And not equal to each other. Let $a>b>c>0,x>y>z>0$ $$\begin{cases} a+b+c=x+y+z\\ abc=xyz \end{cases}$$ now I have try out $1+8+... 3answers 2k views ### Find three real orthogonal matrices of order$3$having all integer entries. [closed] Find three real orthogonal matrices of order$3$having all integer entries. I have no idea to solve the problem. I don't know how to start. If$A$be such matrix then$AA^T=A^TA=I_3$. Please help me.... 1answer 113 views ### Finding integral points on an elliptic$y^2-3y=x^3+x^2\$ curve using the LMFDB-database
I have the following elliptic curve that I want to look up in the LMFDB-database: $$\text{k}:\space\space\space y^2-3y=x^3+x^2$$ Using the Weierstrass form of my elliptic curve, I wrote my equation ...