query: Prove that for odd $n > 1$ , $3^{n} + 1$ is not divisible by $n$
positive_passages.docid: 32166
positive_passages.text: Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$
negative_passages.docid: 148455
negative_passages.text: Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
negative_passages.docid: 74523
negative_passages.text: Proof by Cases involving divisibility
negative_passages.docid: 145558
negative_passages.text: Prove that $2^n+3^n $ is never a perfect square
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: 122019
negative_passages.text: Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
negative_passages.docid: 128306
negative_passages.text: Show that for all $n ∈ Z$, $2n^3 + 3n^2 + n$ is divisible by both 2 and 3.
negative_passages.docid: 2522
negative_passages.text: Why is $a^n - b^n$ divisible by $a-b$?
negative_passages.docid: 27063
negative_passages.text: Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
negative_passages.docid: 129238
negative_passages.text: Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.
negative_passages.docid: 66578
negative_passages.text: Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
negative_passages.docid: 35666
negative_passages.text: How to prove the divisibility rule for $3\, $ [casting out threes]
negative_passages.docid: 17660
negative_passages.text: Show that the odd prime divisors of $n^2+n+1$ are of form $6k+1$ (exclude 3)
negative_passages.docid: 52644
negative_passages.text: Prove $2^k+1$ divisible by 3 for odd K
negative_passages.docid: 68145
negative_passages.text: If $p$ is an odd prime and $k$ an integer with $0<k<p-1$ then $1^k + 2^k + \ldots + (p-1)^k$ is divisible by $p$
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: 135892
negative_passages.text: Understanding the proof for showing that there are infinitely many $n\in\mathbb{Z}^+$ such that $n!+1$ is divisible by at least two distinct primes