# 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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### Is there any way to predict the largest number of consecutive quadratic or cubic residues modulo prime $p$?

We all know that $a$ is a quadratic residue modulo $p$ if and only if $a^{(p-1)/2} \equiv 1 \pmod p$, also $a$ is a cubic residue modulo $p$ if and only if $a^{(p-1)/3} \equiv 1 \pmod p$. Now, for a ...
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### Are primes of the form $6k+1$ a cube modulo $n$, if $3\nmid n$ and none of the prime factors of $n$ is of the form $6k+ 1$?

I wonder if we can assume the following statement to be true in general: Let $p$ be a prime of the form $6k+1$ and $n<p$ a natural number less than $p$. If $3$ does not divide $n$ and none of the ...
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### Are cube roots evenly distributed modulo primes?

Are the cube roots of two integers chosen from a uniform distribution between $1$ and $p-1$ inclusive, $p$ prime, essentially evenly distributed? Note that I will use a $p$ such that $p$ is not ...
• 6,037
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### Applications of higher order reciprocity laws

I'm currently studying quartic & cubic residues and their reciprocity laws, and would like to know of any real world applications to finding the values of their respective residue symbols. I ...
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1 vote
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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 ...
• 8,027
1 vote
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### $n$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 $\left(\dfrac{a}{p}\right)_n=\zeta$...
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### 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 ...
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### 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 ...
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### $\sqrt{2}+1$ 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 $\mathbb{F}_{p^2}$ ...
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