query: Prove by induction: $\forall n \in \mathbb{n}, n \geq 1 \Rightarrow 2^{2^{n}}-1$ is divisible by at least n distinct primes
positive_passages.docid: 2147
positive_passages.text: Different ways to prove there are infinitely many primes?
negative_passages.docid: 71650
negative_passages.text: Proving by strong induction for a sequence of integers, $2^n$ divides term $n$
negative_passages.docid: 131771
negative_passages.text: Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
negative_passages.docid: 75329
negative_passages.text: Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$
negative_passages.docid: 153123
negative_passages.text: Proving by induction $4^n  0$
negative_passages.docid: 31292
negative_passages.text: Divisibility Proof with Induction - Stuck on Induction Step
negative_passages.docid: 130093
negative_passages.text: Prove by Induction : $n < 2^n$
negative_passages.docid: 116518
negative_passages.text: Proof by induction: $n$th Fibonacci number is at most $ 2^n$
negative_passages.docid: 31074
negative_passages.text: Mathematical Induction divisibility $8\mid 3^{2n}-1$
negative_passages.docid: 82774
negative_passages.text: Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
negative_passages.docid: 30374
negative_passages.text: If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$
negative_passages.docid: 40692
negative_passages.text: Proving by strong induction that $\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $
negative_passages.docid: 139269
negative_passages.text: Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction
negative_passages.docid: 154475
negative_passages.text: Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
negative_passages.docid: 96332
negative_passages.text: Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
negative_passages.docid: 101927
negative_passages.text: Induction: Prove that $4^{n+1}+5^{2 n - 1}$ is divisible by 21 for all $n \geq 1$.