query: Let $n \in \Bbb N$. Let $p>2$ a prime number. Show that $1^n+2^n+...+(p-1)^n \equiv 0 \pmod {p}$
positive_passages.docid: 74898
positive_passages.text: $1^n +2^n + \cdots +(p-1)^n \mod p =$?
negative_passages.docid: 94917
negative_passages.text: Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
negative_passages.docid: 88036
negative_passages.text: Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
negative_passages.docid: 12210
negative_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
negative_passages.docid: 131385
negative_passages.text: How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
negative_passages.docid: 154305
negative_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
negative_passages.docid: 74512
negative_passages.text: How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 51382
negative_passages.text: Proving that either $2^n-1 $ or $ 2^n+1$ is not prime
negative_passages.docid: 149508
negative_passages.text: Prove $9^n \equiv 9 \pmod{24}\quad \forall n \in \Bbb N^*$
negative_passages.docid: 144851
negative_passages.text: For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
negative_passages.docid: 2802
negative_passages.text: If $2^n+1$ is prime, why must $n$ be a power of $2$?
negative_passages.docid: 103732
negative_passages.text: Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
negative_passages.docid: 70066
negative_passages.text: Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
negative_passages.docid: 113763
negative_passages.text: Let $p$,$\ q$ be prime numbers and $n\in \mathbb{N}$ such that $p \nmid n-1$. If $p \mid n^q-1$, then $q \mid p-1$.
negative_passages.docid: 38727
negative_passages.text: Modular Exponentiation and Primes
negative_passages.docid: 66851
negative_passages.text: If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$