query: Find all integer $n$ such that $x^2+x+1$ divides $X^{2n}+(x+1)^{2n}+1$
positive_passages.docid: 21439
positive_passages.text: For which $k\in \Bbb N$ does $x^2+x+1$ divide $ (x + 1)^{2k}+x^{2k} + 1$?
negative_passages.docid: 135768
negative_passages.text: Find $P(n+1)$ for a polynomial P
negative_passages.docid: 13120
negative_passages.text: Find all integer solutions to $x^2+y^2+z^2=2xyz$
negative_passages.docid: 87379
negative_passages.text: How to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction?
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
negative_passages.docid: 108382
negative_passages.text: Prove $x^n-1$ is divisible by $x-1$ by induction
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 80907
negative_passages.text: Primes dividing a polynomial
negative_passages.docid: 55437
negative_passages.text: When does $2^{n}$ divide $3^{n}-1$
negative_passages.docid: 118538
negative_passages.text: Find all integers $x$ such that $x^2+3x+24$ is a perfect square.
negative_passages.docid: 44906
negative_passages.text: Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$
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: 65826
negative_passages.text: Find a prime number $p$ and an integer $b<p$ such that $p$ divides $b^{p−1}−1$.
negative_passages.docid: 69136
negative_passages.text: factorization of a^n+1?
negative_passages.docid: 115244
negative_passages.text: Integer solutions of $m^2+(m+1)^2=n^4+(n+1)^4$
negative_passages.docid: 110695
negative_passages.text: Divisor of $2^{2^n} + 1$ modulus $2^{n+1}$