query: let p be a prime number such that {p = 3 [4]} show that the equation {x ^ 2 + 1} = {[p]} does not admit solutions in Z
positive_passages.docid: 135465
positive_passages.text: Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
negative_passages.docid: 100255
negative_passages.text: If a prime divides the hypotenuse of a primitive Pythagorean triple, then $p \equiv 1$ (mod 4)
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 50596
negative_passages.text: $p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
negative_passages.docid: 46292
negative_passages.text: Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
negative_passages.docid: 114489
negative_passages.text: Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$
negative_passages.docid: 50781
negative_passages.text: $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
negative_passages.docid: 7085
negative_passages.text: Prove that if the equation $x^{2} \equiv a\pmod{pq}$ has any solutions, then it has four solutions.
negative_passages.docid: 75318
negative_passages.text: How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?
negative_passages.docid: 136310
negative_passages.text: Prime number in a polynomial expression
negative_passages.docid: 38748
negative_passages.text: There are infinitely many prime numbers $p$ for which $p + 2$ and $p + 4$ are also prime numbers
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: 131488
negative_passages.text: Show that $p^3+4$ is prime
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: 50663
negative_passages.text: prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
negative_passages.docid: 75206
negative_passages.text: If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?
negative_passages.docid: 87682
negative_passages.text: Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$