query: Showing that $p$ is a Fermat prime if and only if every quadratic non-residue mod $p$ is also a primitive root mod $p$
positive_passages.docid: 153862
positive_passages.text: A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.
negative_passages.docid: 31064
negative_passages.text: Showing $x^8\equiv 16 \pmod{p}$ is solvable for all primes $p$
negative_passages.docid: 128575
negative_passages.text: Elementary proof that $3$ is a primitive root of a Fermat prime?
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: 70463
negative_passages.text: Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 115241
negative_passages.text: If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$.
negative_passages.docid: 344
negative_passages.text: Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
negative_passages.docid: 51952
negative_passages.text: Show that $x^{n-1}+\cdots +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime.
negative_passages.docid: 70628
negative_passages.text: Prove p-adic valuation is completely additive for prime p
negative_passages.docid: 13478
negative_passages.text: Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
negative_passages.docid: 76644
negative_passages.text: For any odd prime $p$, a quadratic is solvable mod $p^2$ if it is solvable mod $p$, and $p$ does not divide the discriminant and leading coefficient.
negative_passages.docid: 68635
negative_passages.text: When is $2$ a quadratic residue 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: 95594
negative_passages.text: Prove that a polynomial is irreducible or the field contains a $p$th root
negative_passages.docid: 94017
negative_passages.text: If p is a prime and a ∈Z. How do we prove that the congruence x^x ≡ a (mod p) is solvable?
negative_passages.docid: 34323
negative_passages.text: Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$