An integer $a$ is a quadratic residue modulo $p$ if $a \equiv x^2 \pmod p$ for some integer $x$. If $a$ is not a quadratic residue modulo $p$, it is said to be a quadratic nonresidue modulo $p$.

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### Proving that if prime p>3 divides a^2 + 12, then p is congruent 2(mod 3)

Prove that if prime $p>3$ divides $a^2 + 12$, then $p$ is congruent $2\pmod 3$. I tried splitting (-12/P) to (-1/P) * (3/P) and solving 4 different cases to find when (-12/P) = 1, but I got that p ...
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### Quadratic residue test for mod powers of 2

For odd primes, you can test using Euler's criteria if a number is a Quadratic Residue $\bmod p$. I am looking for a test for mod powers of 2 (which are even & hence cannot use Euler's criteria). ...
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### What is the most complete binary quadratic form table?

Is there a free, easily accessible quadratic form resource — maybe even just a big web-accessible table — which shows “all known results” [reasonably speaking] about numbers of the form $mx^2+ny^2$ ...
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### Question about Legendre Symbol and Division Q12(a) [closed]

Let $q>3$ be an odd prime and let $q=2Q+1$. Prove that: (a) $q\,|\,(3^Q-1) \iff q\equiv ±1 \pmod {12}$ adpated from Number Theory with Applications, James A. Anderson, James M. Bell. Ch 3.9 Q12 (a) ...
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### Showing $\sum_{m=1}^{q-1}m^{(q+1)/2}\equiv0\bmod{q}$, where $q$ is an odd prime congruent to $3 \bmod{4}$

I'm reading through Davenport's "Multiplicative Number Theory", and came across this expression on page 53. $$\sum_{m=1}^{q-1} m^{(q+1)/2} \equiv 0 \mod q,$$ where $q$ is an odd prime ...
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### An exercise in “Number Theory” by Shafarevich and Borevich [duplicate]

I have trouble in solving a basic exercise of the book Number Theory by Shafarevich and Borevich. It is exercise 4, chapter 1, page 4 in my edition. It goes as follows: Using the properties of the ...
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### Number of Quadratic residues less than $p/3$

Suppose that $p\equiv 1\mod 4$ is a prime. I am interested in getting a formula for the number of quadratic residues less than $p/3$. The interest is because there are $(p-1)/4$ quadratic residues ...
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### Perfect squares and cubes in quadratic number fields

Suppose we are given a quadratic number field $\mathbb{Q}(\sqrt{d})$, for some integer $d$ which is not a perfect square. I wish to study when is an element $\alpha \in \mathbb{Q}(\sqrt{d})$ a perfect ...
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### Hasse-Minkowski principle and square theorems

This is a question in the same spirit than this one, trying to prove algebraic number theoretic statements from zeta functions. I want to prove the Hasse-Minkowski principle for quadratic forms in two ...
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### When is $x$ a square in $\mathbb{F}_p[x]/(Q)$?

Let $p$ be a prime and $Q$ be an irreducible polynomial over $\mathbb{F}_p[x]$. Which are all $p$ and $Q$ such that there exists a polynomial $R(x) \in \mathbb{F}_p[x]$ such that $Q$ divides $x - R^2$?...
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### Find all solutions of $x^2 \equiv 9 \pmod{85}$

I am asked to solve this problem, and I know how to solve congruences of degree $2$ modulo a prime $p$, but note that $85=5\cdot 17$ is a product of two primes. On the fly I managed to rewrite the ...
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### Prove that $x^4 \equiv a^2 \pmod p$ is solvable for all $a$

Let $p$ be a prime such that $p \equiv 3 \pmod{4}$ Prove that $x^4 \equiv a^2 \pmod p$ is solvable for all $a$. I'm asked to work out this problem and I'm wondering if my approach is correct? My ...
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### Why does $a^n \mod p$ always result in a number with Legendre symbol as 1?

I noticed that the following expression $a^n\mod p$ where p is a prime and $n >=1$ and $n <= p$ always results in a number with Legendre Symbol (with p as the prime) as 1. I tested it out with a ...
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### Formula for the root of $x^2=-3\mod n$ when $n=p_1^{k_1}\cdots p_l^{k_l}$ and $p_i$ primes equal to $1\mod 3$

Consider the equation $$x^2=-3\mod n,$$ where $n=p_1^{k_1}\cdots p_l^{k_l}$ with $p_i$ primes equal to $1\mod 3$. Notice that any such $n$ can written as $a^2+3b^2$ (by a theorem of Fermat) for some ...
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### Solve $x^2 \equiv -1 \pmod{41}$ [duplicate]

Using Legendre's symbol, knowing that $41 \equiv 1 \pmod{4}$ we can solve said equation. Then, I looked at the residue classes and I just counted my way to the solution $x = 9$. Verifying, $9^2 = 81$ ...
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### For odd $n$, if $x^2 = a \pmod{n}$ has a solution then at least one solution has a specific representation.
Update: ̶F̶o̶r̶ ̶o̶d̶d̶ ̶$n$,̶ ̶i̶f̶ ̶ $x^2 = a \pmod{n}$ ̶h̶a̶s̶ ̶a̶ ̶s̶o̶l̶u̶t̶i̶o̶n̶ ̶t̶h̶e̶n̶ ̶a̶t̶ ̶l̶e̶a̶s̶t̶ ̶o̶n̶e̶ ̶s̶o̶l̶u̶t̶i̶o̶n̶ ̶h̶a̶s̶ ̶a̶ ̶s̶p̶e̶c̶i̶f̶i̶c̶ ...