Questions tagged [quadratic-reciprocity]
The quadratic-reciprocity tag has no usage guidance.
28 questions
6
votes
2
answers
425
views
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
I already posted this question on MSE.
Using theorem $IV$ from this article, it is possible to prove that when $p$ is a prime such that $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the ...
- 387
11
votes
2
answers
615
views
Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
- 17.9k
7
votes
1
answer
339
views
Rational prime factors in the components of powers of Gaussian primes
Let $\pi=a+bi\in \mathbb{Z}[i]$ be a Gaussian prime with $a$ and $b$ nonzero, and $b$ even. For odd rational primes $p=\pi\bar\pi$ and $q\neq p$, define $\pi^{\frac{1}{2}\left(q-\left(\frac{-1}{q}\...
1
vote
1
answer
260
views
Conceptual explanation for extra/missing $p$ solutions to $x^2+y^2=a \pmod p$ at $a=0$
Throughout, $p$ will denote a prime integer, and $k$ an arbitrary integer.
I have worked through V. Lebesgue's proof of quadratic reciprocity outlined by Keith Conrad in this MO thread, and I feel ...
- 833
1
vote
0
answers
136
views
Quadratic equations over Gaussian integers
Given an equation $x^2\equiv(a+ib)\bmod(c+id)$ where $a,b,c,d\in\mathbb Z$ holds, how to test if the equation has solutions and how to find the solutions in polynomial in $\log(|abcd|)$ time if $c+id$ ...
- 13.9k
7
votes
2
answers
781
views
Could a nice principle be extracted from this lemma of Gauss
I asked the following question in the math SE, with a bounty of 200 pts, without result.
question:
To prove the quadratic reciprocity law, Gauss needed the following lemma:
If $p$ is a prime number ...
- 687
12
votes
1
answer
856
views
Quadratic reciprocity for three primes?
The quadratic reciprocity law states that for $p_1\ne p_2$ prime, the product $\left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)$ takes values $1$ or $-1$ depending on whether $p_1$ and $p_2$ ...
- 23k
1
vote
0
answers
477
views
Legendre Symbol of a Very, Very Large Value
I'm trying to use FLINT (Fast Library for Number Theory) to calculate the Legendre Symbol of the following:
$$\left(\frac{n! + 1}{p}\right)$$
In my case, $p$ is a positive, odd prime (specifically $...
- 119
7
votes
0
answers
326
views
Expressing quartic Dirichlet characters modulo primes $p\equiv 1\bmod 4$ with Legendre symbols
Looking through some old notes of mine from two years ago I found some crude notes writing what amounted to the statement that for any prime $p\equiv 1\bmod 4$ one could express for any odd integer $p\...
- 2,506
8
votes
0
answers
237
views
Connection between Gauss's lemma and Zolotarev's lemma
So I was reflecting on the relationship between Gauss's Lemma and Zolotarev's Lemma in proofs of quadratic reciprocity:
GL: $(a/p) = -1^n$, where $n$ is the number of least positive residues of $ax$ ...
- 595
2
votes
1
answer
128
views
Density of "simultaneous squares"
Let $(u,v)$ be a pair of non-zero integers. We say that $(u,v)$ is a pair of simultaneous squares if for all primes $p$ dividing $u$, we have $\left(\frac{v}{p}\right) = 1$ and for all primes $q$ ...
- 26.9k
36
votes
3
answers
2k
views
Nonabelian reciprocity law
I heard the following relation in a talk by Peter Scholze. Could someone explain "in a simple way" what is the precise relation between the polynomial $x^4-7x^2-3x+1 $ and the integral ...
- 461
7
votes
0
answers
326
views
Chowla's Construction of prime having least quadratic non-residue $\gg \log p$
This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the
first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues".
I recently ...
- 306
3
votes
1
answer
294
views
Given n and q, how to find p so q$\neq$n-th power (mod p)?
Reasonable exceptions allowed on $q$. Example solution: $n=2$.
Suppose $q$ is odd. Let $p$ be so $pq\equiv -1$ (mod 8). Then $q\neq$ 2nd power (mod $p$) is the same as $\left(\frac{q}{p}\right)=-1$...
- 197
1
vote
1
answer
215
views
Computation of Hilbert symbol of order 4
We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for the ...
- 13
11
votes
2
answers
1k
views
distribution of $\sqrt{-1} \mod p$
While reading up on quadratic reciprocity, I learned that if $p = 4k+1$ then $-1$ has a square root in $\mathbb{Z} / p \mathbb{Z}$.
Let $r_p$ be an integer with $0\leq r_p < p$ and $r_p^2 \equiv -...
- 22.8k
3
votes
0
answers
165
views
Averages of $L(s,\chi)$
Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol.
What is the
abscissa of convergence
of the double Dirichlet series ?
$$
\sum_{\substack{m,n \in \mathbb{N} \\ \gcd(m,n)=1 \\m,n\equiv 1 \...
- 3,062
8
votes
1
answer
458
views
higher reciprocity theorems from ratios of Gauss sums
One recent proof of quadratic reciprocity involves computing various rations of the Gauss sum.
In Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation Gurevich, ...
- 22.8k
8
votes
1
answer
1k
views
Biquadratic reciprocity for $p\equiv 1\pmod 4$ and $q\equiv 3\pmod 4$
For a prime $p\equiv 1\pmod 4$, let $\left(\frac{\cdot}{p}\right)_4$ denote the rational biquadratic residue symbol; that is,
$$ \left(\frac{a}{p}\right)_4 =
\begin{cases}
\ \ \ ...
- 23k
9
votes
1
answer
2k
views
Consecutive non-quadratic residues
Inspired by this recent question, I wondered if a similar result is true for quadratic non-residues, namely, if it is true that for every $k \in \mathbb{N}$ there exists a prime $p$ such that exists $...
- 499
21
votes
1
answer
2k
views
Gauss linking integral and quadratic reciprocity
In the setting of Mazur's "primes and knots" analogy, prime ideals $\mathfrak p\subset\mathcal O_K$ correspond to "knots" $\operatorname{Spec}\mathcal O_K/\mathfrak p$ inside a "3-manifold" $\...
- 18.7k
24
votes
2
answers
2k
views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
- 26k
66
votes
5
answers
6k
views
What do theta functions have to do with quadratic reciprocity?
The theta function is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the ...
- 1,063
14
votes
3
answers
3k
views
Quadratic reciprocity and Weil reciprocity theorem
I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...
- 5,063
11
votes
1
answer
2k
views
Weil reciprocity vs Artin reciprocity
This is probably an easy question for the experts:
Given two rational functions $f$, $g$ on a non-singular projective algebraic curve X (over an algebraically closed field $k$) and $p \in X$, one ...
- 893
24
votes
1
answer
2k
views
Can Eisenstein's lattice point proof of quadratic reciprocity be generalized?
I'm referring to this proof. The key formula ("Eisenstein's Lemma") is
$$\left(\frac{q}{p}\right)=(-1)^{\sum_{u}\lfloor\frac{qu}{p}\rfloor},\text{ where $u=2,4,\ldots,p-1$}$$
The sum in the ...
- 6,792
21
votes
2
answers
2k
views
Context for "Coronidis Loco" from Weil's Basic Number Theory
In Samuel James Patterson's article titled Gauss Sums in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Patterson says
"Hecke [proved] a beautiful theorem on the different ...
- 7,062
117
votes
22
answers
39k
views
What's the "best" proof of quadratic reciprocity?
For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.