query: Choose a positive number $k > 1$ so that $b^k \equiv a \pmod{91}$
positive_passages.docid: 49938
positive_passages.text: Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
negative_passages.docid: 95033
negative_passages.text: Find the remainder of a number with a large exponent
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 16660
negative_passages.text: Prove that any positive number has an $n$th root
negative_passages.docid: 63661
negative_passages.text: integers $n$ such that $(n-1)!+1$ is a power of $n$
negative_passages.docid: 154933
negative_passages.text: Prove $(p-k)!(k-1)!\equiv (-1)^k \text{ mod p }$
negative_passages.docid: 119046
negative_passages.text: Expressing a positive integer as a sum of positive integers
negative_passages.docid: 133819
negative_passages.text: How to prove that $364 \mid n^{91} - n^7$?
negative_passages.docid: 17601
negative_passages.text: How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
negative_passages.docid: 126354
negative_passages.text: Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$
negative_passages.docid: 111505
negative_passages.text: Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
negative_passages.docid: 56574
negative_passages.text: For any integer $a,b$ let $N_{a,b}$ denote the number of positive integer $x<1000$ satisfying $x= a( mod\;27)$ and $x=b(mod\;37)$. Then
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
negative_passages.docid: 152870
negative_passages.text: Near integers in powers of binomials with radicals
negative_passages.docid: 134352
negative_passages.text: The number of possible factorizations of a positive integer.
negative_passages.docid: 156377
negative_passages.text: Set one or more of coefficients to a specific integer
negative_passages.docid: 21874
negative_passages.text: Solve $x^3+x \equiv 1 \pmod p$