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Questions tagged [cubic-reciprocity]

Use this tag for questions about theorems in number theory that state conditions under which the congruence x³ ≡ p (mod q) is solvable.

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Quadratic and Cubic Fractional Residues

Is there any method to find out the characterization of all primes $p$ such that $\frac{a}{b}$ is a quadratic residue modulo $p$ such that $a$ and $b$ are primes? Is there any method to do the same ...
60 views

$n$th power residue conjecture

Let $\left(\dfrac{a}{p}\right)_n$ be the $n$th power residue symbol such that when $a$ is an $n$th power residue $\pmod p$, $\left(\dfrac{a}{p}\right)_n=1$ otherwise $\left(\dfrac{a}{p}\right)_n=\zeta$...
185 views

Quintic reciprocity conjecture

Let $p=x^4 + 25x^2 + 125$ be a prime. Prove that $2$ is a quintic residue $\pmod p$, and therefore $y^5=2\pmod p$ is solvable. A similar example was first conjectured by Euler: If $p=x^2 + 27$ is a ...
107 views

For what values of n is $f(x) = x^3 mod(n)$ a bijection from $X={(0,1,2,…,n-1)}$ to itself.

I was thinking about shuffling, mapping ${(1,2,...,n-1)}$ to a permutation of itself using a mapping like $x\to x^k \mod(n)$ and clearly $k=2$ cannot work since $1$ and $n-1$ have the same image for ...
84 views

$\sqrt{2}+1$ is a cube in $\mathbb{F}_{p^2}$ when…

I have a conjecture and I think I have a class field theory proof of it, but I would like to know if there's a QR or CR proof of it. The statement is that $\sqrt{2}+1$ is a cube in $\mathbb{F}_{p^2}$ ...