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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### How can one go about determining which primes $p>3$ satisfy the Jacobi symbol $(\frac{-3}{p})=1$? [closed]

I know how to do this for a positive 3, but the -3 is throwing me off a little. I’m not exactly sure how to proceed with determining the primes to satisfy that Jacobi symbol. Thanks for any help 84 views

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### Enumerating odd Quadratic Residues modulo $p, p^m, n$

I has asked this question recently and this came up as a followup question. Q1. Is there a way to efficiently enumerate odd quadratic residues less than $a$ modulo $p$? I know we can use quadratic ...
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### Sign of fundamental unit in real quadratic number fields with 1 mod 4 discriminant factors

Let $K$ be a real quadratic number field of discriminant $D$ with fundamental unit $\varepsilon$. Further, I want to assume that each positive factor $n$ of $D$ satisfies $n \equiv 1 \pmod 4$. (For ...
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### What's the maximum order of $A \in SL_2(\mathbb{Z}/p\mathbb{Z})$ if $tr(A)+2$ and $tr(A)-2$ are both either quadratic residue modp or non quad. res??

I have already proved that $SL_2(\mathbb{Z}/p\mathbb{Z})$ is a group with the product of matrices. I also proved that $|SL_2(\mathbb{Z}/p\mathbb{Z})|=p^3-p$ using the first isomorphism theorem and the ... 107 views

### Find all primes $p$ such that $3x^2-1\equiv 0 \mod{p}$ has a solution

So I generally know how to do this for equations where $x^2$'s coefficient is $1$. Completing squares and then using quadratic reciprocity I can find that primes for which there are solutions depend ...
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Gauss' well-known law of quadratic reciprocity states the following: Main Theorem. Given two disttinct primes, $p$ and $q$, if at least one of $p-1$ and $q-1$ is divisable by $4$, then the congruency ...
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### Problem regarding quadratic recidues that I cannot solve [duplicate]

The problem is as follows: Let $p$ be a prime and assume that $p=1$ mod 5. Let $c\in\left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}$ be an element of order 5 and let $g=2(c+c^{-1})+1$ Show that $g^2=5$ ...
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### Finding roots of a polynomial using quadratic reciprocity

Does the polynomial $X^2− X + 19$ have a root in $\mathbb Z/61\mathbb Z$? I am unsure of how to go about this problem but I outlined the way I have been approaching these problems in the problem below....
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### If $p\equiv 1 \;\text{mod}\; 3$, then show that one can find an integer $k$ satisfying $k^2-k+1=p\cdot M\;$ with $M<p$

If $p\equiv 1 \;\text{mod}\; 3$, then show that one can find an integer $k$ satisfying $k^2-k+1=p\cdot M\;$ with $M<p$ ($p$ is a prime) I don't have any clue on how to work this problem. Also, if ...
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### When is $-3$ a quadratic residue mod $p$?

Going over a past exam in my elementary number theory course, I noticed this question that caught my attention. The question asked for the conditions that allowed $-3$ to be a quadratic residue mod $p$...
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### Are $3$ and $19$ the only primes representable by the principal form with discriminant $-57$?

I'm trying to see if there are other primes, but so far I only managed to get $3$ and $19$ by factoring $57$. How would I find other primes, if they do indeed exist?
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### Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$
$\textbf{Question:}$Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$ I could easily see that the given is equivalent to showing that there are infinitely many primes $p$ ...