# Tagged Questions

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

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### $a^b+2$ or $a^b-2$ is in set

Let $A$ be an infinite set of positive integers. For any two $a,b\in A$, $a\neq b$, at least one of the numbers $a^b+2$ and $a^b-2$ are also in $A$. Must $A$ contain a composite number?
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### New Identities for Generalized Fibonacci Numbers?

Over the past few months I have been investigating one the generalizations of the Fibonacci numbers, called the Generalized Fibonacci Numbers (GFNs). The GFNs are just like the regular Fibonacci ...
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### Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
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### Repeated application of interesting function on tuples

This question was inspired by Thursday's CIMC. Suppose you have a function $$f_n: (\Bbb{Z}/n\Bbb{Z})^n\to(\Bbb{Z}/n\Bbb{Z})^n; (a_1,a_2,a_3,\dots,a_n)\mapsto (b_1,b_2,b_3,\dots,b_n)$$ defined as ...
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### $Ax+By+Cz=D \text { has a solution iff } \gcd(\gcd(A,B),C)\mid D$ [duplicate]

I read today that $Ax+By+Cz=D \text { has a solution iff } \gcd(\gcd(A,B),C\mid D$ but I can't find it again, I also can't find any Diophantine equations with 3 variables that doesn't have solutions ...
### How to solve the following equation in $\mathbb{Z}_n$?
Given an $n\in\mathbb{N}$, $a\in \mathbb{Z}_n$ and $x,y\in\mathbb{Z}$, how do I approach to solving the following equation: $a^x \equiv a^y \mod n$ I think that from here I can deduce that: \$x \...