# Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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### rewriting one expression as other expression

Can anyone explain as how we can rewrite the first expression as second ? I am not able to pick the step done to change from 1 to 2. ...
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### $\left(\frac{3}{p} \right)=1$ iff $p\equiv 1\pmod{12}$ or $p\equiv -1\pmod{12}$

Let $p\geq 5$ be a prime. $\left(\frac{3}{p} \right)=1$ iff $p\equiv 1\pmod{12}$ or $p\equiv -1\pmod{12}$. So $\left(\frac{3}{p} \right)=\left(\frac{p}{3} \right)\cdot (-1)^{(p-1)/2}$ and this ...
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### Find the Wrong Student

There are 15 student in the class and each of them has a different number 1 to 15. Student #1: wrote the natural number on the board. Student #2 said : This number is divisible by my number(number ...
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### a particular linear combination

Fix $a_1,\ldots,a_n\in\mathbb{N}$. I'd like to know if one can characterize the natural numbers that belong to the set $$\{b_1a_1+\ldots+b_na_n:\,b_j\in\{-1,0,1\}\}.$$ EDIT: Maybe this question doesn'...
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### Colors on sets $S=\{1,2 \cdots ,1000\}$.

To each element of sets $S=\{1,2 \cdots ,1000\}$ a color is assigned. Suppose that for any two elements $a$ and $b$, of $S$,if $15$ divides $a+b$, then they both are assigned with same color. What is ...
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### Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
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### $\Bbb Z[\sqrt{-5}]$ is not a PID [duplicate]

I want to show: In a PID $R$ two elements $a,b\in R$ always have a greatest common divisor. Therefore $\Bbb Z[\sqrt{-5}]$ is not a PID. For the first part: $I=\{ax+by:x,y\in R\}$ is an ideal, ...
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### Infinitely many primes $\equiv 3 \mod 4$

Question 1 Is the following proof of the infinitude of primes $\equiv 1\mod 4$ okay? Consider a prime divisor $p\mid (n!)^2+1$. Then $(n!)^2\equiv -1 \mod p$, hence $n!$ has multiplicative order $4$ ...
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### Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
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### Proof that $n = 3k + 5l$ for $n > 7$

Show that for every n greater than $7$, there are non-negative integers $k$ and $l$ such that $$n = 3k+ 5l.$$ So induction seems like a possibility. $n = 3k + 5l$ and so $n + 1 = 3k + 5l + 1$. ...
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### Non-negative integer solutions to $4ab-a-b=c^2$

The puzzle is as follows: Problem: Find all non-negative integer solutions to $4ab-a-b=c^2$ My Progress: There is, of course, the trivial solution of $a=b=c=0$, and I suspect there are no more (...
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### Find the gcd of the following Gaussian integers

$\gcd(5 + 8i, 3 + 2i)$ in $Z[i]$. I found it and I got 1 then I look at the manual solution and it turns out it can be i or -i or -1 or 1. why?
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### Finite summation including binomial coefficients and double factorials

I came across the following summation: $$\sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}).$$ $\tbinom{n}{k}$ are binomial coefficients, $n!/k!(n-k)!$. Mathematica told ...
### If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 - \frac{5}{3q}$?
Let $\sigma=\sigma_{1}$ be the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q$ satisfies $q \equiv k \equiv 1 \pmod 4$), and $k=1$, does ...