query: How to prove that $2^{2n + 1} - 9n^2 + 3n - 2$ is divisible by $54$ for $n \in \mathbb{N} \cup \{0\}$?
positive_passages.docid: 150462
positive_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: 122386
negative_passages.text: How do I prove that $3$ divides $14^{2n}-1$ by induction?
negative_passages.docid: 68561
negative_passages.text: How to show $\lceil ( \sqrt3 +1)^{2n}\rceil$ where $n \in \mathbb{ N}$ is divisible by $2^{n+1}$
negative_passages.docid: 104965
negative_passages.text: Find $n$ such that polynomial is divisible
negative_passages.docid: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 122019
negative_passages.text: Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
negative_passages.docid: 17194
negative_passages.text: $6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$
negative_passages.docid: 6943
negative_passages.text: Idea behind proof that $\frac{21n+4}{14n+3}$ is irreducible for all $n$
negative_passages.docid: 15348
negative_passages.text: prime divisor of $3n+2$ proof
negative_passages.docid: 9302
negative_passages.text: Questions about Divisibility of $2^n$ by $3$
negative_passages.docid: 113359
negative_passages.text: Proving that $x^2 + 4$ is not divisible by $3$
negative_passages.docid: 127833
negative_passages.text: Proving that $7^n(3n+1)-1$ is divisible by 9
negative_passages.docid: 33011
negative_passages.text: Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
negative_passages.docid: 35666
negative_passages.text: How to prove the divisibility rule for $3\, $ [casting out threes]
negative_passages.docid: 31074
negative_passages.text: Mathematical Induction divisibility $8\mid 3^{2n}-1$
negative_passages.docid: 63416
negative_passages.text: Prove $(n^5-n)$ is divisible by 5 by induction.
negative_passages.docid: 75733
negative_passages.text: Divisibility of prime numbers