query: Using Fermat's Little Theorem, prove $1^{p-1} + 2^{p-1} + 3^{p-1} +\ldots+(p-1)^{p-1} \equiv -1 \pmod p$
positive_passages.docid: 21629
positive_passages.text: Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
negative_passages.docid: 112390
negative_passages.text: Solutions of $x^2 \equiv 1 \pmod N$
negative_passages.docid: 128575
negative_passages.text: Elementary proof that $3$ is a primitive root of a Fermat prime?
negative_passages.docid: 87267
negative_passages.text: Conjecture similar to Fermat's Theorem.
negative_passages.docid: 20980
negative_passages.text: Prove that $\frac{p^p-1}{p-1}$ is not prime if $p \equiv 1 \pmod 4$
negative_passages.docid: 44729
negative_passages.text: Prove $g$ is a generator if $g^q=1 \pmod p$
negative_passages.docid: 41673
negative_passages.text: Solve congruence system $\ x\equiv m_i-1 \pmod{m_i}\,$ for $\,i = 1,\ldots, k$
negative_passages.docid: 113986
negative_passages.text: Fermat's Last Theorem: implications (there is no new proof)
negative_passages.docid: 37093
negative_passages.text: If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots  \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
negative_passages.docid: 35167
negative_passages.text: Prove congruence using fermat's thm
negative_passages.docid: 68233
negative_passages.text: Fermat's little theorem's proof for a negative integer
negative_passages.docid: 34323
negative_passages.text: Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
negative_passages.docid: 2218
negative_passages.text: Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
negative_passages.docid: 38805
negative_passages.text: Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
negative_passages.docid: 121441
negative_passages.text: $1\cdot3\cdot5 \cdots (p - 2) = (-1)^{m+k+1} \pmod p$, and $2\cdot4\cdot6\cdots(p - 1) = ( -1)^{m +k} \pmod p$.
negative_passages.docid: 46147
negative_passages.text: Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
negative_passages.docid: 54286
negative_passages.text: Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$