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Questions tagged [quadratic-reciprocity]

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. (Ref: http://en.m.wikipedia.org/wiki/Quadratic_reciprocity)

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Reference: Wikipedia

The theorem was conjectured by Euler and Legendre and first proven by Gauss. He refers to it as the "fundamental theorem" in the "Disquisitiones Arithmeticae".

272 questions
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Quadratic Reciprocity: Determine if 11 is a quadratic residue $\mod p $for primes of the form: 44k+5?

I'm currently studying for exams and this has me stuck. A sample question from a past paper states: Use the quadratic reciprocity theorem to determine whether $11$ is a quadratic residue $\mod p$ for primes of the form: (i) …
David Ryan
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Technological incompetence

I am a high school student going into my junior year. Earlier this summer, I went to a math camp at Ohio State University, and one of the assignments was to prove quadratic reciprocity. I was told that one of the proofs involved lattice points, so…
George Bentley
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Find $m, n$ such that $\frac{n^2 + 1}{m^2 + 1 }$ is an integer multiple of a perfect square

I'm trying to find $n,m \in \mathbb{N}$ such that $\sqrt{ \frac{n^2+1}{2(m^2+1)}}$ is rational. I see that if $a,b$ are relatively prime $\sqrt{ \frac{a}{b}}$ is rational if and only if $a,b$ are perfect squares. $n^2+1$ can be a perfect square…
scibuff
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Quadratic reciprocity for odd integers

I have a book that proves the three laws of quadratic reciprocity for primes i.e $\left(\frac{-1}p\right)=1$ if $p=1\mod4$ and $-1$ if $p=3\mod4$, where $p$ is prime, etc. But in the end the book says that the three laws are also true for arbitary…
TheGeometer
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Legendre symbols (p/q) = (a/q)

Suppose q and p are odd primes and p = q+4a for some integer a. From the properties of the Legendre symbol and the Law of Quadratic Reciprocity prove that the following two identities hold. (p/q) = (a/q), and (a/p) = (a/q) So i first did the LAW,…
comewithme
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Quadratic character of 3

Using the QRL prove that, for any odd prime $p$, $(3/p) = 1$ if $p$ is congruent to $1$ or $11 \pmod{12}$. Using the Quadratic reciprocity law, $(3/p)(p/3)=(-1)^{(3-1)(p-1)/4}$, I get that the Legendre symbol is always equal to $1$. What am I…
Christian
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Prove that there exists a number $x$ such that $x^2 \equiv 2$ (mod $p$) and $x^2 \equiv 3$ (mod $q$)

Let $p$ and $q$ be distinct odd primes for which $(2/p)$ and $(3/q)$ are both $1$. Prove that there exists a number $x$ such that $x^2 ≡ 2$ (mod $p$) and $x^2 ≡ 3$ (mod $q$). This is my attempt to solve it: $x^2=2$ (mod $p$) and $y^2=3$ (mod $q$) $p…
Sohail Abdul Karim
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