query: Does there exist a positive integer $n$ such that $4^n - 1$ is not divisble by $3$?
positive_passages.docid: 2522
positive_passages.text: Why is $a^n - b^n$ divisible by $a-b$?
negative_passages.docid: 27428
negative_passages.text: Divisibility test for $4$
negative_passages.docid: 34784
negative_passages.text: Is a number meeting these conditions divisible by forty-nine?
negative_passages.docid: 117167
negative_passages.text: $3^{n+1}$ divides $2^{3^n}+1$
negative_passages.docid: 131108
negative_passages.text: Can $3p^4-3p^2+1$ be square number?
negative_passages.docid: 52542
negative_passages.text: For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?
negative_passages.docid: 101349
negative_passages.text: $-1$ isn't a quadratic residue $\pmod{p}, p=4k+3$.
negative_passages.docid: 23427
negative_passages.text: How to prove there are no more positive integers that are products of 2 and 3 consecutive numbers?
negative_passages.docid: 116540
negative_passages.text: Positive integers less than 1000 without repeated digits
negative_passages.docid: 21906
negative_passages.text: Show $3^m + 3^n +1$ cannot be a perfect square for $m,n$ being positive integers.
negative_passages.docid: 152172
negative_passages.text: Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
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: 67291
negative_passages.text: Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
negative_passages.docid: 32653
negative_passages.text: Smallest integer divisible by all up to $n$
negative_passages.docid: 513
negative_passages.text: If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.
negative_passages.docid: 71935
negative_passages.text: Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.
negative_passages.docid: 8034
negative_passages.text: Can $n!$ be a perfect square when $n$ is an integer greater than $1$?