query: If $a^n-1$ divides $b^n-1$ too often, then $b=a^k$
positive_passages.docid: 97021
positive_passages.text: Do there exist two primes $p<q$ such that $p^n-1\mid q^n-1$ for infinitely many $n$?
negative_passages.docid: 72511
negative_passages.text: If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
negative_passages.docid: 108187
negative_passages.text: When does $p^2$ divide $an^k + bp$?
negative_passages.docid: 8366
negative_passages.text: Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd
negative_passages.docid: 9319
negative_passages.text: If $n\ne 4$ is composite, then $n$ divides $(n-1)!$.
negative_passages.docid: 47060
negative_passages.text: Split $n$ into nontrivial factors via a nontrivial square-root of $1\!\pmod{\!n}$
negative_passages.docid: 12694
negative_passages.text: Why does $a^n - b^n$ never divide $a^n + b^n$?
negative_passages.docid: 158106
negative_passages.text: Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime
negative_passages.docid: 24098
negative_passages.text: Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
negative_passages.docid: 45469
negative_passages.text: If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$
negative_passages.docid: 133905
negative_passages.text: Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
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: 88995
negative_passages.text: Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
negative_passages.docid: 117761
negative_passages.text: When is $\binom{n}{k}$ divisible by $n$?
negative_passages.docid: 99908
negative_passages.text: When does $120$ divide $n^5 - n$
negative_passages.docid: 111505
negative_passages.text: Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
negative_passages.docid: 62982
negative_passages.text: Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$