# 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)

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### Is there a proof of quadratic reciprocity using $p$-adic numbers?

I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can get it by considering the cyclotomic extension of $\mathbb{Q}_{p}$. ...
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### Proof that $x^4 - qy^4 = az^2$ has no integral solution

This is a question from Takashi Ono's book, Problem 1.45 to be exact. The question is Let $q$ be a prime such that $q = 1 \mod 8$ and $a$ be an integer such that $p^2\not\mid a$ for any prime $p$ and ...
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### if $19a^2 \equiv b^2 \pmod 7$ then $19a^2 \equiv b^2 \pmod {7^2}$

I am stuck with this problem. All what I can tell is that $19a^2 \equiv 5a^2 \equiv b^2 \pmod 7$ and $5$ is not a quadratic residue$\pmod 7$. Any hints please,,
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### Solutions to $x^2+x-1\equiv 0$ mod $p$

The problem is to find all prime number p such that the above congruence has solutions. I started this problem by rearranging the equation such that: $$x(x+1)\equiv 1 \pmod{p}$$ The hint given was ...
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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 ...
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### Elementary Number Theory: Quadratic Reciprocity

Note that $2717 = 11*13*19$ and determine if $x^2 \equiv 295$ (mod $2717$) is solvable. I know I have to spilt this up into three different congruences $x^2 \equiv 295$ (mod $11$), $x^2 \equiv 295$ (...
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### Prove $\sum\limits_{j=1}^{p-1} j\left(\frac{j}{p}\right) = 0$ for an odd prime $p$ with $p\equiv 1\text{ mod } 4$

I want to show for an odd prime $p$ with $p\equiv 1\text{ mod } 4$, that $$\sum\limits_{j=1}^{p-1} j\left(\frac{j}{p}\right) = 0$$ where $\left(\frac{j}{p}\right)$ is the Jacobi symbol. I got ...
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### how to determine $x^2 \equiv a \pmod m$ solvable if m,a not coprime

I am reading books about Number theory now. It's hard to me. Seems I use quadratic residues to determine if $x^2 \equiv a\pmod m$, but I don't know how when m, a not coprime. More specifically, like: ...
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### Legendre/Jacobi symbol, reciprocity laws etc.

Is there a good textbook that will help me understand the motivation for defining the Legendre symbol (and it's Jacobi generalization), and applications of them to number theory? I have a math degree ...
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### Quadratic residues in finite field

For an integer $a$ and a finite field $F_{q}$ of odd order, what is the efficient algorithm to determine $a$ is Quadratic residue or not?
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### For which primes $p$ is $-3\in Q_p$?

I'm really bad at doing these for some reason, just need some help, this is not a hw question. I just need to do these smaller problems to gain some understanding. For which primes $p$ is $-3\in Q_p$ ...
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### $n^6$ does not divide $6^n$

Find the number of positive integers $n=2^a3^b\, \, (a,b\geq 0)$ such that $n^6$ doesn't divide $6^n$. I have not encountered such types of question before and so I donot know how to even approach ...
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### Using Gauss's Lemma to determine primes of a quadratic residue

The problem I'm attempting to solve is: Apply Gauss's Lemma to determine the primes of which -2 is a quadratic residue. I'm completely lost.
### Use the Law of Quadratic Reciprocity and the Chinese Remainder Theorem to determine when $6$ is a quadratic residue modulo $p$.
I'm working on the following problems concerning quadratic reciprocity. a) Use the Law of Quadratic Reciprocity and the Chinese Remainder Theorem to determine for which primes $p$, $-50$ is a ...