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)

244 questions
Filter by
Sorted by
Tagged with
31 views

74 views

### Question on proving Quadratic Reciprocity with Gauss sums

I'm trying to understand the proof of Quadratic Reciprocity on Wikipedia (this link: https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity#Proof_using_algebraic_number_theory, then the section ...
50 views

98 views

### Primes of the form $p=X^2+3Y^2$

I'm trying to work in $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ to show that an odd prime $p\in\mathbb{Z}$, $p\neq 3$ is of the form $p=X^2+3Y^2$ if and only if $p\equiv1$ mod $3$. The hint is to ...
39 views

### Gaussian sets modulo $p$.

I have the following problem, given the definition: Let $p$ be an odd prime, and let $S$ be a set of $(p-1)/2$ integers. We call $S$ gaussian set modulo $p$ if $S\cup-S=S\cup\{-s\mid s\in S\}$ Is a ...
40 views

### Is there an archive or a collection of quadratic reciprocity proofs?

After Pythagorean theorem the quadratic reciprocity has the largest amount of proofs, I heard there are more than 160 proofs of quadratic reciprocity. I have seen a few of these proofs but I was ...
83 views

### Proving first supplement of the quadratic reciprocity law [duplicate]

I'm trying to prove the first supplement of the quadratic reciprocity law in its particular form: $$-\bar{1} \text{is a square in} \mathbb{Z}/p\mathbb{Z} \iff p \equiv 1 \pmod 4$$ For the forward part,...
77 views

### Prove a formula to evaluate Legendre Symbol $(\frac{-3}{p})$ using quadratic Gauss Sums (i.e. without using quadratic reciprocity)

I've been trying to find a question similar to this, but have failed to find any. The full statement is let $\zeta = e^{\frac{2\pi i}{3}}$, and use the fact that $(2 \zeta + 1)^2 = -3$ and quadratic ...
69 views

### Use the quadratic reciprocity to show that 5 is a square modulo $M_l$ if and only if $l$ ≡ 1 (mod 4).

Let $l$ be an odd prime such that $M_l = 2^l − 1$ is a Mersenne prime. Use the quadratic reciprocity to show that 5 is a square modulo $M_l$ if and only if $l$ ≡ 1 (mod 4). Here's what I know: Using ...
49 views

### Is it possible to derive the quadratic reciprocity law from the decomposition law in a quadratic extension?

How can I derive the quadratic reciprocity law from the decomposition law for a prime in a quadratic extension? There is no need to read the following texts. I know the necessary and sufficient ...
62 views

53 views

### Law of Quadratic Reciprocity for primes that are congruent 3 or 1 modulo 4

I have a question regarding some claims I read about the Law of Quadratic Reciprocity that I can't fully understand. The law itself is written as follows: For all odd numbers $P,Q \in \mathbb{N}$ with ...
81 views

### 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 ...
68 views

### 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 ...
85 views

### 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....
73 views

### 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 ...
306 views

### 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 $p$...
52 views

### 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?
45 views

141 views

### 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 $p$ ...
70 views

### 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^...
57 views

### 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 ...
43 views

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. ...
50 views

28 views

### 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 ...
51 views

### 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 $p>2$. ...
60 views

115 views

### 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 ...
59 views

### "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 ...
104 views

### 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?
73 views

### 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 ...
65 views

### How to show $\,2^n \equiv 8\pmod{24}\,$ for odd $\,n\ge 3$?

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 ...
44 views

### 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. ...
46 views

### 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 ...
45 views

### 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 ...