# Questions tagged [divisibility]

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

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### Approaching a Divisibility Question

Perhaps I'm a bit rusty on my number theory, but I have been struggling to figure out how to determine what positive integers $x$ and $y$ satisfy $$x+y \text{ divides } x^2 + y^2$$ Any thoughts? I ...
1 vote
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### $(a^m+b^m)\mid(a^n+b^n) \iff m\mid n$ [duplicate]

Prove that $(a^m+b^m)\mid(a^n+b^n) \iff m\mid n$. Here $a, b, m, n\in\mathbb{Z}^+$, $m\leq n$ and $(a, b)=1$. This is a questions from a number theory book that I am recently studying. I have read ...
1 vote
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### Infinitely many $a_n$ [closed]

Let $\{a_n \}$ be a sequence defined as follows: $$\begin{gathered} a_0=0 ; a_1=1 ; a_2=2 \text { and } \\ a_{n+3}=5^n \cdot a_{n+2}+n^2 \cdot a_{n+1}+11 a_n \text { for } n \geq 0 . \end{gathered}$$...
1 vote
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### Another Binomial Coefficient Congruence Modulo Prime Powers

I have the following conjecture on binomials modulo prime powers: Let $s, b, n \ge 0$ be integers, let $p$ be a prime and let $0\le k_0 < p^{b+1}$ and $0\le n_0 < p^{b+s}$, then we have the ...
1 vote
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### Formulas that generate "too few" primes

I am looking for examples of formulas $f: \mathbf{N} \rightarrow \mathbf{N}$ for which the number of conjectured primes $p \in \text{ Im}(f)$ is either finite or less than what naive heuristics would ...
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### Prove that a*b is not divisible by prime number p [duplicate]

Can't understand if my thoughts make sense or not. The question is the following: $ab$ is not divisible by prime $p$, if both $a$ and $b$ are not divisible by $p$. I thought to prove it like this: For ...
1 vote
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### Given $a,b \in \mathbb{N}$, is it true that $\phi(\gcd(a,b))=\gcd(\phi(a),\phi(b))$?

Let $a,b \in \mathbb{N}$ and $d=\gcd(a,b)$. Then we have that $a/d$ and $b/d$ are coprime. I am trying to see if $\phi(a)/\phi(d)$ and $\phi(b)/\phi(d)$ are also coprime ($\phi$ is the Euler totient ...
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### Find N such that for all x $\ge$ N, $P(x)$ is Divisible by Primes Bigger than 10

Given an arbitrary distinct natural numbers, $d_1, d_2, d_3, d_4,$ and $d_5$. Let $P(x) = (x + d_1)(x + d_2)(x + d_3)(x + d_4)(x + d_5)$. Prove that there is a number $N$ (in terms of $d_1, d_2, ...$...
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### The exact meaning of $1/0$?

What is the exact meaning of $1/0$? Does that mean a number that is very large, a number that cannot be expressed as the one we know, infinite numbers of number so that giving one particular value is ...