# Tagged Questions

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

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### Need help with the proof of a theorem about Gaussian integers

Theorem 6-3. If $\alpha$ and $\beta$ are integers of $Z[i]$, and $\beta \neq 0$ then there are $\kappa$ and $\rho$ in $Z[i]$ such that $$\alpha =\beta\kappa+\rho, \text{ } N_\rho < N_\beta$$ ...
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### How can be proven that any number X is greater,lesser or equal to any other number Y?

I have looked for it on the internet, really, but all I have found are particular cases like 1 > 0, or such. Is there an algebraic proof for proving that x > y or, x = y, or x < y? I thought of ...
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### If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
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### Smallest chain of consecutive integers not all coprime

Let $t$ be a positive integer. What is the smallest $t$ for which we can find an integer $a$ such that each element of the set $\{a+1,a+2,\dots ,a+t\}$ is not coprime with all other elements of the ...
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### lower bound for sum of distinct n-th roots of unity

Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$ Let us consider the set$S = \{ |s(\vec x)| : \vec x \...
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### Can it proved that the GCD does not divide the integer coefficients in the linear form of the GCD?

Let $d = (a,b)$ then $d = ax +by$ for some $x,y \in \mathbb{Z}$ I want to prove that $d \nmid x,y$. Motivation I'm trying to solve the following problem: If $a$ is prime to $b$ and $y$, $b$ is ...
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### Showing that $2^6$ divides $3^{2264}-3^{104}$

Show that $3^{2264}-3^{104}$ is divisible by $2^6$. My attempt: Let $n=2263$. Since $a^{\phi(n)}\equiv 1 \pmod n$ and $$\phi(n)=(31-1)(73-1)=2264 -104$$ we conclude that $3^{2264}-3^{104}$ is ...
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### Generalization of Inkeri's primality test

How to prove that following hypothesis is true ? Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are ...
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