query: For any integer a, a$^{37}$ $\equiv$ a (mod 1729)
positive_passages.docid: 12210
positive_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
negative_passages.docid: 94855
negative_passages.text: Prove if a (mod n) has a multiplicative inverse, then it's unique
negative_passages.docid: 96326
negative_passages.text: Number of $q$-th residues modulo $n$
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 78416
negative_passages.text: Show that $7$ divides $a^{15}-a^3$ for any integer $a$
negative_passages.docid: 95653
negative_passages.text: Euler's theorem: [3]^2014^2014 mod 98
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: 43940
negative_passages.text: Showing that $n$ is pseudoprime to the base $a$
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: 51153
negative_passages.text: Number of solutions to $x^2\equiv a\pmod m$
negative_passages.docid: 40968
negative_passages.text: show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
negative_passages.docid: 48268
negative_passages.text: How many $\alpha \in S_n$ are such that $\alpha^2 = 1$?
negative_passages.docid: 4972
negative_passages.text: Mod of numbers with large exponents [modular order reduction]
negative_passages.docid: 4129
negative_passages.text: Solvability of $a \equiv x^2 \mod b$
negative_passages.docid: 32803
negative_passages.text: Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)
negative_passages.docid: 38876
negative_passages.text: $r! \equiv (−1)^k \pmod p$
negative_passages.docid: 17601
negative_passages.text: How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$