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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### Let $p=40k+9$ be prime. Does $10$ always have even order mod $p$?

This came up while answering a question on the period of the decimal expansion of $1/p$. The critical factor was whether the period (aka the order of $10$ mod $p$) is even or odd, equivalently ...
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### Show for any odd prime $p\geq 5,$ $(-3/p)=1$ or $-1$ [duplicate]

Show for any odd prime $$p\geq 5,$$ $$\left ( \frac{-3}{p} \right ) =\begin{cases} 1 & \text{ if } p\equiv 1,-5\pmod{12} \\ -1& \text{ if } p\equiv -1,5\pmod{12} \end{cases}$$ So far I have ...
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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 $p$ ...
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### Using quadratic residues and/or reciprocity to prove relative primality?

I have odd positive integers $q$ and $y$, with $3q^2 < y^2 < 4q^2$, such that the following are true: \begin{align} (q^2+9) &\mid (y^2+5)(y^2+29) \\[0.25em] (q^2+2) &\mid (y^2+1)(y^...
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### Trial Division and Quadratic Reciprocity

I am reading The Joy of Factoring by Samuel Wagstaff and I am having trouble understanding a paragraph from this book. It says the following One can use quadratic residues to speed Trial Division ...
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How can we prove there are infinitely many solutions to $\frac{1}{x^{2}-2x+3}=y$ by only staying at Further maths at High School level? Will the graph ever go below the x-axis or will stay on it. ...
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### Show $\chi_{\pi}\left(a(-1)^\frac{p-1}{4}\right)=(-1)^\frac{a^2-1}{8}.$

I'm currently going through of Ireland and Rosen's 'A Classical Introduction to Modern Number Theory' and would like some help on CH9Ex29. I know that $a(-1)^\frac{p-1}{4}$ is primary (i.e it is ...
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### Applications of higher order reciprocity laws

I'm currently studying quartic & cubic residues and their reciprocity laws, and would like to know of any real world applications to finding the values of their respective residue symbols. I ...
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### A number theory problem: show $N_{x^2−n}(p^k) = 2$ for all $k = 1,2,3,…$

One of my number theory exercises this week asks the following: Let $n$ be an odd natural number and assume that the Legendre symbol $\left(\frac np\right)$ equals $1$ for some prime $p>2$. ...
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### How to evaluate $\prod_{n=1}^{p-1} \sin(\frac{2n^2\pi}{p})$

According to Quadratic Gauss sum I want to know what is the exact value of this product?,since I put this product on Wolfram Alpha and I got the result in this form $$\frac{a+b\sqrt{ p}}{c}$$ for some ...
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### “Inverting” the Artin map in terms of characters

I will start by saying that I know very little about Class Field Theory, so I am hoping for someone to shed some light on this. If $K$ is a finite abelian extension of $\mathbb{Q}$, then one can ...
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### Assume that p and q are odd primes and $p \equiv q \pmod {28}$. Show that $(\frac{7}{p}) = (\frac{7}{q})$.

Assume that p and q are odd primes and $p \equiv q \pmod {28}$. Show that $(\frac{7}{p}) = (\frac{7}{q})$. Could anyone give me a hint for the solution please?
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### Why if the Legendre symbol satisfy $\left(\frac{a}p\right)=\left(\frac{p}a\right)$ then $\left(\frac{a}p\right) = 1$?

Sorry for my stupid question: This is in completion to this question Let $p$ be a prime of the form $p = a^2 + b^2$ with $a,b \in \mathbb{Z}$ and $a$ an odd prime. Prove that $(a/p) =1$ Why if the ...
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### Exponentiation with odd number in modular arithmetic

Show that if $n\geq3$ is odd, then $2^n-1\equiv7\mod24$. I tried solving this backwardly. We want to prove that $2^3(2^{n-3}-1)=2^n-2^3\equiv0\mod24$. Since $\frac{24}{2^3}=3$, this leaves us to ...
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### Question on modular arithmetic with prime numbers

Let $p\neq3$ be a prime conguent to $3\pmod4$ and let $q$ be a prime divisor of $(12p)^{2019}+1$ satisfying $q\equiv p^2+1\pmod{3p}$. Determine $q\pmod 4$. I tried solving the problem as follows. ...
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### Proving that $n$ is not a prime [duplicate]

Let $n=3^{100}+2$ and assume that $X^2-53$ does not have zeroes in $\mathbb{Z}/ n\mathbb{Z}$. Show that $n$ is not a prime. I tried solving this problem by assuming that $n$ is a prime (in order to ...
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### Solution to equation modulo p

Under the assumptions that $$p\cong 1 \mod 5$$ and $$g = 2(c+c^{-1})+1$$ where $c$ has order $5$ modulo $p$. I need to show that $g^2 \cong 5 \mod p$. I have that $$g^2=4(c^4+c^3+c^2+c)+9$$ I know ...
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### Quadratic reciprocity in Langlands program

I know quadratic reciprocity is the easiest example of langlands correspondence. Langlands correspondence gives some relation between automorphic forms and artin representations. My question is: what ...
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### If $d$ divides $a^4+a^3+2a^2-4a+3$, prove that $d$ is a fourth power modulo $13$

If $d$ divides $f(a)=a^4+a^3+2a^2-4a+3$, prove that $d$ is a fourth power modulo $13$. $f(a)\equiv{(a-3)}^4\pmod {13}$. But how can we prove any divisor of $f(a)$ is a fourth power? If we prove that ...
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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 ...
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 $11$), $x^2 \equiv 295$ (...