query: prove that $a^{N-1} \not\equiv 1(\bmod N)$
positive_passages.docid: 2904
positive_passages.text: Solving linear congruence (modular inverse or fraction) via gcd Bezout equation
negative_passages.docid: 76495
negative_passages.text: How many points in RSA, such that $m^e = m \bmod n$
negative_passages.docid: 28136
negative_passages.text: $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$
negative_passages.docid: 29231
negative_passages.text: Proof of : $x^{n}\equiv x^{\varphi(m)+[n \bmod \varphi(m)]} \mod m$
negative_passages.docid: 93820
negative_passages.text: Fermat's little theorem applied for $n$'th root existence $\!\bmod p$
negative_passages.docid: 40663
negative_passages.text: $a_1 a_1', a_2 a_2', ..., a_n a_n'$ is not a reduced modulo system $\mod p$.
negative_passages.docid: 108237
negative_passages.text: Show $a^p \equiv b^p \mod p^2$
negative_passages.docid: 46741
negative_passages.text: Prove that $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer without using concepts of abstract algebra
negative_passages.docid: 43246
negative_passages.text: Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
negative_passages.docid: 61860
negative_passages.text: Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$.
negative_passages.docid: 12636
negative_passages.text: Product Rule for mod operator: $ ab\bmod p = ((a\bmod p)(b\bmod p))\bmod p$
negative_passages.docid: 15314
negative_passages.text: Number of solutions to $x^n \equiv 1 \mod p$
negative_passages.docid: 36747
negative_passages.text: $\text{Tor}_0^R(M,N)\simeq M\otimes_R N$
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: 79219
negative_passages.text: Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
negative_passages.docid: 125868
negative_passages.text: Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
negative_passages.docid: 12210
negative_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$