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A Conjecture on The Generalization of Quadratic Reciprocity Law

Is there any way to prove the following conjecture regarding the Generalization of Quadratic Reciprocity Law. The statement being, $$ \left(\dfrac{a_1}{a_2}\right)\left(\dfrac{a_2}{a_3}\right) ...
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1answer
20 views

Exercise of Quadratic Reciprocity

This is an exercise in Burton : Prove that $$(5/p) =1\ iff\ p\equiv 1,\ 9,\ 11,\ or\ 19\ (20) $$ Note that $5=4+1$ so that $(5/p)=(p/5)$. In further $$ (p/5)^2=(5/p)(p/5) = (-1)^{1\cdot ...
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0answers
29 views

Find a criterion for the primes p such that (5/p) = 1. [duplicate]

Find a criterion for the primes p such that (5/p) = 1. I don't understand this question it is like Determine all primes P such that (5/p)=1 I appreciate any help
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3answers
111 views

Elementary, direct proof of when $5$ is a quadratic residue mod $p$

$\newcommand{\kron}[2]{\left( \frac{#1}{#2} \right)}$ It's easy to use Quadratic Reciprocity to show that $\kron{5}{p} = \kron{p}{5} = 1$ when $p \equiv \pm 1 \pmod 5$, and is $-1$ when $p \equiv \pm ...
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0answers
24 views

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 ...
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1answer
26 views

finding all primes $p$ for which a given number is a quadratic residue

I have seen an exercise on the Apostol, but I haven't understood some passages. I would be very grateful if you could solve my doubts. The problem is Find all primes $p$ for which 3 is a ...
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0answers
20 views

For which positive odd integers $n,$ $(n,33)=1$ does the Jacobi symbol $\left(\frac{33}{n}\right) = 1$?

Attention: What modulus should we use to insure that n is odd? So I know that $\left(\frac{33}{n}\right) = \left(\frac{11}{n}\right)\left(\frac{3}{n}\right)$ then using the legendre properties ...
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2answers
68 views

factor 9997 using quadratic reciprocity

I looked up the factorization and it is $13\cdot769$, but I have no idea how quadratic reciprocity allows you to deduce this without knowing it. I thought maybe ...
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1answer
323 views

Is every integer a quadratic residue mod some p?

Is every integer (say $d$) a quadratic residue mod some prime number $p$?
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0answers
35 views

When can a congruence relation be transformed into quadratic reciprocity expressions?

When can a congruence relation $$p \equiv c_1, c_2, \ldots, c_r \mod{N}$$ be transformed back into quadratic reciprocity expressions $$\left (\frac{d_1}{p} \right) = \left (\frac{d_1}{p} \right) = ...
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2answers
44 views

Let p be an odd prime, q the smallest quadratic non residue (mod p). Prove q is prime.

So I have this problem; Let p be an odd prime and let q be the smallest positive integer which is a quadratic non residue (mod p). Prove q is a prime. So what I know is that, since q is the ...
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1answer
135 views

Quadratic reciprocity problem

How can I use quadratic reciprocity to prove that $-3$ is a quadratic residue $\pmod p$ if and only if $p=2$ or $p \equiv 1 \pmod 6$ and deduce that $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_p ...
0
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1answer
40 views

Quadratic congruences problem $(x^2-a)(x^2-b)(x^2-ab)$ ≡ 0 (mod p)

Let p be an odd prime and let a,b ∈ Z such that p ∤ a. Prove that the congruence $(x^2-a)(x^2-b)(x^2-ab)$ ≡ 0 (mod p) is always solvable. Not sure where to begin here.
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2answers
152 views

elementary proof that infinite primes quadratic residue modulo $p$

$p \gt 2$ is a prime, then there are infinite primes $q$ such that $q$ is a quadratic residue modulo $p$. With Dirichlet's theorem on arithmetic progressions, the problem is easy! How about ...
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1answer
55 views

Analytic proof of quadratic reciprocity

Is there any proof of quadratic reciprocity that is more analytic than those described on Wikipedia (http://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity)?
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1answer
88 views

Consecutive quadratic residues

I was studying quadratic reciprocity laws and came across the following question: Is it true that for every $k \in \mathbb{N} $ there exists a prime $p$ such that $1,2,...,k$ are all quadratic ...
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1answer
41 views

Quadratic Reciprocity as a consequence of Eisenstein Reciprocity

I was recently looking at the wikipedia page on Eisenstein Reciprocity, which says it "extends Quadratic Reciprocity." However, though the two do seem to be related, I don't completely understand how ...
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1answer
70 views

Prove that S consists of the quadratic residues mod (p) and T consists of the quadratic non-residues mod (p)

All the followings are $\bmod$ $p$ Let p be an odd prime. Suppose that the set $X = \{ 1,2, . . . , p-1\}$ can be written as the union of two nonempty subsets $S$ and $T$, where $S \neq T$, such that ...
2
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1answer
183 views

Quadratic Reciprocity Joke

I found a joke on a site made by user Zev Chonoles on quadratic reciprocity, the joke is as follows: $$\text{Quadratic reciprocity: } \left(\frac{p}{q}\right)=\left(\frac{q}{p}\right), \text{ up to ...
2
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1answer
32 views

Jacobi symbol $\left(\frac{(n+1)/2}{n}\right)$

Let $(\frac{a}{n})_J$ be Jacobi symbol defined by \begin{equation} \left(\frac{a}{n}\right)_J=\left(\frac{a}{p_1}\right)^{e_1}\left(\frac{a}{p_2}\right)^{e_2}\cdots\left(\frac{a}{p_k}\right)^{e_k} ...
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2answers
38 views

Number of elements that exist so $b^3 \equiv a\pmod n$, when n is composed of p and q who are prime numbers

Given $2$ prime numbers,$ p$ and $q$, that are both not even, and $3$ doesn't divide $p-1$ or $q-1$, and $n=pq$, how many elements in $Z^*_n$ exists that has $b$ such that $b^3\equiv a\pmod n$ . I ...
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6answers
335 views

What is the point of quadratic residues?

What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most ...
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1answer
70 views

Legendre symbols (p/q) = (a/q) [duplicate]

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) = ...
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2answers
52 views

$2^{4n+1} \equiv 1 \pmod{8n+7}$, this has been bugging me

Here is the question: Suppose that $p$ is an odd prime. The law of quadratic reciprocity says that $x^2\equiv 2\pmod p$ has a solution. if $p\equiv1 \text{ or } 7 \pmod 8$. Prove that ...
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2answers
77 views

If $x^2 \equiv a \bmod p$ has a solution, does it necessarily have infinitely many solutions? [closed]

If not, how would you prove it for a specific case? Such as, for example, $x^2 \equiv 2 \bmod 7$ ?
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1answer
48 views

Suppose $p$ and $q$ are odd primes and $p = q + 4a$ for some $a$. Prove that $(\frac ap) = (\frac aq)$ holds. [duplicate]

This problem also had me prove that $(\frac pq) = (\frac aq)$, but I've already managed to do that. I've tried messing around with the Law of Quadratic Reciprocity but can't get anything. I've also ...
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2answers
28 views

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 ...
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1answer
38 views

Prove that $\left(\frac2p\right) = 1$ if $p \equiv 1,7 \pmod 8$ and $\left(\frac2p\right) = -1$ if $p \equiv 3,5 \pmod 8$ using ring theory

Let $p$ be an odd prime number and let $\alpha = [X] \in R=\mathbb F_p[X]/\langle X^4+1\rangle$, and $y = \alpha + \alpha^{-1}$ I've proven: 1) $\alpha$ is a primitive eight root of unity in $R$. ...
0
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1answer
147 views

Quadratic Reciprocity [duplicate]

For any integer $a$ and any two primes $p,q$ with $(p,q)=1$. Prove that if $p \equiv q\pmod {4a}$, then $\displaystyle\left(\frac ap\right)=\left(\frac aq\right)$ I know I need to write $a=(−1)^{e_0} ...
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1answer
115 views

Quadratic Reciprocity

For any integer $a$ and any two primes $p,q$ with $(p,q)=1$. Prove that if $p \equiv q$ mod $4a$, then $(\frac{a}{p})=(\frac{a}{q})$ I know I need to write $a=(−1)^{e_0} 2^{e_2} p_1 p_2 \cdots p_r$ ...
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0answers
40 views

Show that $x^2\equiv a \pmod {2^n}$ has a solution where $a\equiv1 \pmod 8$ and $n\ge3$

Show that $$x^2\equiv a \pmod {2^n}$$ has a solution where $a\equiv1 \pmod 8$ and $n\ge3$ Actually, the question I had to solve was more complicated something like this: $x^2\equiv a ...
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2answers
136 views

Show that prime $p=4n+1$ is a divisor of $n^{n}-1$

Show that the prime number $p=4n+1$ is a divisor of $n^{n}-1$ Ok, the question itself is simple as hell, but I couldn't think of a simple way to solve this question. I tried to solve the ...
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0answers
131 views

When quadratic formula $ax^2 + bx +c = 0$ (mod p) is solvable modulo prime p?

I am learning about quadratic congruences and I don't now how to decide, for which a,b,c and p there is a solution of the congruence. It is sufficient if the determinant $\sqrt{b^2-4ac}$ has a ...
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0answers
35 views

How do I prove that $x^2\equiv-1\pmod{p}$ [duplicate]

How do I prove that $x^2\equiv-1\pmod{p}$ iff $p$ is prime at form: $p=4n+1$. I have to use Wilson theroem... (I'm asking this, becuase I didn't understand it from my prev. Q, how it proves that ...
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1answer
349 views

Solve quadratic congruence using the Chinese Remainder Theorem

Solve $x^2 \equiv$ -1 (mod 205) I have that as 205 = 5 x 41 this becomes: $x^2$ $\equiv $ -1 (mod 5) $x^2$ $\equiv $ -1 (mod 41) then i'm not sure where to go. I can see 32 is a solution but I ...
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2answers
114 views

Number Theory: Solutions of $ax^2+by^2\equiv1 \pmod p$

Assume $p$ is a prime number and $\gcd(ab, p)=1$. Show that the number of integer solutions $(x, y)$ of $ax^2+by^2 \equiv 1 \pmod p$ is $$p - \left(\dfrac{-ab}{p}\right)$$ where ...
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1answer
46 views

Proving something with Wilson's Theorem [continued.]

At first I asked this: Proving something with Wilson theorem. Now I have to prove that if $p=4n+3$ it's impossible to represent $-1$ in the form $x^2$ modulo $p$. How can I prove it? Thank you!
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2answers
118 views

Proving something with Wilson theorem

I need to prove that $x^2\equiv -1\pmod p$ if $p=4n+1$. ($p$ is prime of course...) I need to use Wilson theorem.
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1answer
141 views

A condition for an odd prime to be represented by a binary quadratic form of a given discriminant

Let $f = ax^2 + bxy + cy^2$ be an integral binary quadratic form. We say $D = b^2 - 4ac$ is the discriminant of $f$. If $D < 0$ and $a > 0$, we say $f$ is positive definite. It is easy to see ...
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1answer
608 views

Proofs of the properties of Jacobi symbol

The definition and properties of Jacobi symbol are stated in this article. I don't have a textbook handy containing the proofs of the following properties of Jacobi symbol. It seems to me that not ...
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1answer
119 views

Quadratic Reciprocity - Legendre Symbols

Find the value of $((1\cdot 2)/73)+((2\cdot 3)/73)+...+((71\cdot 72)/73)$. This is based off each fraction being a Legendre Symbol. I tried to find a pattern... but I could't find anything. Also, I ...
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0answers
79 views

What should be taught about the reciprocity law for high school gifted student

This autumn I have to teach a mini course for a small group of high school student (mathematical gifted class) on Quadratic Reciprocity Law (3 lectures in ten hours). This is the requirement of the ...
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1answer
170 views

Quadratic extensions of $\mathbb Q$

This is a question from Lang's ANT, Thm 6, ch.IV, $\S2$. It states that every quadratic extension of $\mathbb{Q}$ is contained in a cyclotomic extension and that it's a direct consequence of the ...
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1answer
66 views

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 ...
2
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1answer
120 views

Quadratic congruences

Is there an algorithm to solve the quadratic congruence $x^2\equiv D \pmod m$ for any $D$ and $m$? I searched a bit and found algorithms for $m$ prime and $\gcd(D, m) = 1$. None of them gave a ...
2
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1answer
76 views

Quadratic residues mod $n$ of $n-1$

While investigating something about square numbers, I started to wonder which numbers $n$ have quadratic residues of $n-1$. Obviously all $n$ of the form $m^2+1$ will work. But there are other numbers ...
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1answer
49 views

Showing that if $p_1\equiv p_2 \pmod{4n}$, then $(\frac{n}{p_1})=(\frac{n}{p2})$

Let $n$ be an integer and $p_1$, $p_2$ be primes such that $p_1 \equiv p_2 \pmod{4n}$. Prove $\left(\dfrac{n}{p_1}\right)=\left(\dfrac{n}{p_2}\right)$. 2 cases: $p_1=4nk_1+1$, $p_2=4nk_2+1$. ...
3
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2answers
152 views

Probability that $x \equiv 3 \pmod{4}$

I am working on a number theory project and I am interested in the following statement: What is the probability that an integer $x$ has the property that $|x| \equiv 3 \mod{4}$? This seems ...
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2answers
470 views

When computing the Legendre symbol $(\frac{3}{p})$, how to combine different congruences together into one statement?

I try to find Legendre symbol for $\left(\dfrac{3}{p}\right)$. This is what I did so far: case 1: $p=3\mod{4}. $ so $\left(\dfrac{3}{p}\right)=-\left(\dfrac{p}{3}\right)$. now ...
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1answer
64 views

Prove of Quadratic residual

Let $p$ be a prime and $q$ a primitive root modulo $p$. How do I show that q is a quadratic residue modulo $p$ if and only if $a \equiv q^{2k}\pmod p$ for some integer $k$?