query: $q$ an odd prime and $p$ divides $1+n+\dots +n^{q-1}$. Show $p=q$ or $p\equiv 1$ mod $ q$.
positive_passages.docid: 155886
positive_passages.text: let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$
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: 52530
negative_passages.text: There exists a prime congruent to $m$ mod $p$
negative_passages.docid: 64148
negative_passages.text: If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
negative_passages.docid: 15436
negative_passages.text: Find p and q s.t. $({p-1})^{q-1}+({q-1})^{p-1} \equiv 101 \pmod {pq}$
negative_passages.docid: 26503
negative_passages.text: Suppose that $p$ ≥ $q$ ≥ $5$ are both prime numbers. Prove that 24 divides ($p^2 − q^2$)
negative_passages.docid: 54359
negative_passages.text: Odd Primes Problem Proof
negative_passages.docid: 46820
negative_passages.text: If $p$ is a prime such that $p≡1\pmod 7$ then $p ≡1\pmod{14}$
negative_passages.docid: 130653
negative_passages.text: $p^2$ divides $ab$ and $\gcd(a,b)=1$. Show $p^2$ divides $a$ or $p^2$ divides $b$
negative_passages.docid: 14724
negative_passages.text: For primes $p≡3\pmod 4$, prove that $[(p−1)/2]!≡±1\pmod p$.
negative_passages.docid: 27063
negative_passages.text: Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
negative_passages.docid: 110752
negative_passages.text: $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
negative_passages.docid: 513
negative_passages.text: If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.
negative_passages.docid: 140948
negative_passages.text: Suppose $p$ and $q$ are odd primes and $p = q + 4a$ for some $a$. Prove that $(\frac ap) = (\frac aq)$ holds.
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: 153862
negative_passages.text: A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.
negative_passages.docid: 124995
negative_passages.text: $p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$