query: If $(-1)^{\gcd(n,F_n)}=1$ where $F_n$ is the $n$-th Fibonacci number then $n$ is divisible by $6$ ?
positive_passages.docid: 17269
positive_passages.text: Strong Induction Proof: Fibonacci number even if and only if 3 divides index
negative_passages.docid: 82777
negative_passages.text: If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
negative_passages.docid: 97805
negative_passages.text: $n+1$ is a divisor of $\binom{2n} {n}$
negative_passages.docid: 9302
negative_passages.text: Questions about Divisibility of $2^n$ by $3$
negative_passages.docid: 147079
negative_passages.text: $\gcd\left(\binom{n}{k},n\right)>1$ for nontrivial values?
negative_passages.docid: 42435
negative_passages.text: Proof of this result related to Fibonacci numbers: $\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$?
negative_passages.docid: 97947
negative_passages.text: How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$?
negative_passages.docid: 132486
negative_passages.text: Induction proof on Fibonacci sequence: $F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$
negative_passages.docid: 10348
negative_passages.text: Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
negative_passages.docid: 144132
negative_passages.text: The number $2^{3^n}+1$ is divisible by $3^{n+1}$ and not divisible by $3^{n+2}$.
negative_passages.docid: 153892
negative_passages.text: Number Theory divisibilty
negative_passages.docid: 51858
negative_passages.text: Showing $\gcd(2^m-1,2^n+1)=1$
negative_passages.docid: 69279
negative_passages.text: If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
negative_passages.docid: 17194
negative_passages.text: $6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$
negative_passages.docid: 62254
negative_passages.text: Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$
negative_passages.docid: 96535
negative_passages.text: Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
negative_passages.docid: 73391
negative_passages.text: Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$