query: If $(a,b)=1$ and $n$ is a prime, prove that $(a^n+b^n)/(a+b)$ and $(a+b)$ have no common factor, unless $a+b$ is a multiple of $n$
positive_passages.docid: 18714
positive_passages.text: $\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$
negative_passages.docid: 100426
negative_passages.text: Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
negative_passages.docid: 84733
negative_passages.text: Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
negative_passages.docid: 112541
negative_passages.text: Prove that $n^2 + 2^n$ is composite if $n\not\equiv3\pmod{6}$.
negative_passages.docid: 31895
negative_passages.text: Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$
negative_passages.docid: 144622
negative_passages.text: Proof: If no prime less than or equal to $\sqrt{n}$ divides $n$, then $n$ is a prime
negative_passages.docid: 63183
negative_passages.text: Suppose $a,b,m,n$ are natural numbers, prove that if $a({m \over n})=b$ and $(a,n)=1$, then ${m \over n} \in \mathbb{N}$
negative_passages.docid: 43939
negative_passages.text: Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1.
negative_passages.docid: 22845
negative_passages.text: Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
negative_passages.docid: 422
negative_passages.text: If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$
negative_passages.docid: 145041
negative_passages.text: Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
negative_passages.docid: 101159
negative_passages.text: Prove If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n)
negative_passages.docid: 125618
negative_passages.text: Prove $2^{3^n}+1 $ is a multiple of $3^{n+1}$ by induction
negative_passages.docid: 11168
negative_passages.text: Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
negative_passages.docid: 19046
negative_passages.text: Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
negative_passages.docid: 95111
negative_passages.text: Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
negative_passages.docid: 36602
negative_passages.text: $a^m=b^m$ and $a^n=b^n$ imply $a=b$