query: Prove that $2^{3^n}+1$ is divisible by $3^{n+1}$
positive_passages.docid: 117167
positive_passages.text: $3^{n+1}$ divides $2^{3^n}+1$
negative_passages.docid: 66016
negative_passages.text: Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
negative_passages.docid: 46431
negative_passages.text: Proof that $\sum\limits_i \frac{(p-1)!}i$ is divisible by $p$
negative_passages.docid: 151692
negative_passages.text: When does $2^n-1$ divide $3^n-1$?
negative_passages.docid: 38904
negative_passages.text: Prove that $(a + b + c)^{13}$ is divisible by $abc$ if $b|a^3$, $c|b^3$ and $a|c^3$.
negative_passages.docid: 112147
negative_passages.text: Proving $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
negative_passages.docid: 79219
negative_passages.text: Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
negative_passages.docid: 34738
negative_passages.text: Prove that $\gcd(n!+1,(n+1)!+1)=1$
negative_passages.docid: 52032
negative_passages.text: Prove that ${n^5 - n}$ is divisible by 5
negative_passages.docid: 51858
negative_passages.text: Showing $\gcd(2^m-1,2^n+1)=1$
negative_passages.docid: 48503
negative_passages.text: Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
negative_passages.docid: 121706
negative_passages.text: Show $\text{det}(A)$ is divisible by $2^{n-1}$
negative_passages.docid: 150462
negative_passages.text: Prove by induction that for every $n \ge 0$ the term $2^{2n+1} - 9n^2 + 3n -2$ is divisible by $54$
negative_passages.docid: 96268
negative_passages.text: Prove that $5$ divides $3^{3n+1}+2^{n+1}$
negative_passages.docid: 112031
negative_passages.text: Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
negative_passages.docid: 100892
negative_passages.text: Proving divisibility of $a^3 - a$ by $6$
negative_passages.docid: 50894
negative_passages.text: Prove that $6$ divides $n(n + 1)(n + 2)$