The quadratic-reciprocity tag has no wiki summary.
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1answer
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Quadratic residues mod $n$ of $n-1$
While investigating something about square numbers, I started to wonder which numbers $n$ have quadratic residues of $n-1$. Obviously all $n$ of the form $m^2+1$ will work. But there are other numbers ...
1
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1answer
42 views
Showing that if $p_1\equiv p_2 \pmod{4n}$, then $(\frac{n}{p_1})=(\frac{n}{p2})$
Let $n$ be an integer and $p_1$, $p_2$ be primes such that $p_1 \equiv p_2 \pmod{4n}$. Prove $\left(\dfrac{n}{p_1}\right)=\left(\dfrac{n}{p_2}\right)$.
2 cases:
$p_1=4nk_1+1$, $p_2=4nk_2+1$. ...
2
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2answers
103 views
Probability that $x \equiv 3 \pmod{4}$
I am working on a number theory project and I am interested in the following statement:
What is the probability that an integer $x$ has the property that $|x| \equiv 3 \mod{4}$?
This seems ...
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2answers
106 views
When computing the Legendre symbol $(\frac{3}{p})$, how to combine different congruences together into one statement?
I try to find Legendre symbol for $\left(\dfrac{3}{p}\right)$.
This is what I did so far:
case 1: $p=3\mod{4}. $
so $\left(\dfrac{3}{p}\right)=-\left(\dfrac{p}{3}\right)$. now ...
-1
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1answer
39 views
Prove of Quadratic residual
Let $p$ be a prime and $q$ a primitive root modulo $p$. How do I show that q is a quadratic residue modulo $p$ if and only if $a \equiv q^{2k}\pmod p$ for some integer $k$?
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1answer
97 views
If $p \equiv 1 \mod{4}$ is prime, how to find a quadratic nonresidue modulo $p$?
If $p \equiv 3 \mod{4}$ is prime, then $-1$ is a quadratic non-residue modulo $p$. This is not the case when $p \equiv 1 \mod{4}$. How can we find a quadratic non-residue in this case?
At least one ...
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3answers
106 views
Determine the number of solutions for quadratic equation modulo prime
Given that $106$ is quadratic residue $\bmod\ 139$, how can I determine the number of solutions to the following equation?
$$x^2 \equiv 106 \pmod{139}$$
3
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2answers
91 views
Quadratic Reciprocity: Determine if 11 is a quadratic residue $\mod p $for primes of the form: 44k+5?
I'm currently studying for exams and this has me stuck. A sample question from a past paper states:
Use the quadratic reciprocity theorem to determine whether $11$ is a quadratic residue $\mod p$ ...
0
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1answer
108 views
Quadratic reciprocity and proving a number is a primitive root
Let $p$ and $q$ be odd primes. Show that 2 is a primitive root of $q$, if $q = 4p + 1$
1
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1answer
80 views
Law of Quadratic Reciprocity question
If $p$ and $q = 10p+3$ are odd primes, show that the Legendre symbols $(\frac{p}{q})$ and $(\frac{3}{p})$ are equal.
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0answers
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solution count of quadratic form congruences
Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
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3answers
72 views
Is 3 ever a seventh power mod a prime $p$ if $p\equiv 1 (7)$
I had a question asking when is 3 a seventh power modulo a prime $p$ if $p=1(7)$. However, I tried to find just one example using mathematica but I went up to primes in the thousands and I still ...
5
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1answer
61 views
Proving without reciprocity laws that if $p>0$ a prime such $p=1(5)$ then 5 is a quadratic residue mod $p$.
I've done a similar problem where $p=1(3)$ and showed that $-3$ is a quadratic residue modulo $p$. These problems are sledgehammered by reciprocity laws, so I am trying to prove it directly using the ...
2
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2answers
99 views
Where did I go wrong on computing the legendre symbol $\left(\frac{150}{1009}\right)$
Where did I go wrong on computing the legendre symbol $\left(\frac{150}{1009}\right)$.
I know the answer is $1$. For some reason every way I compute this legendre symbol I get $-1$:
...
1
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1answer
102 views
The meaning of the Jacobi Symbol and its efficient evaluation
Are the following jacobi symbol evaluations correct?:
$(\frac{35}{53}) = -1$
$(\frac{68}{233}) = -1$
$(\frac{126}{509}) = 1$
$(\frac{672}{1297}) = 1$
$(\frac{1235}{3499}) = -1$
Also what is the ...
6
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1answer
173 views
Primes of the form $p=a^2-2b^2$.
I've stumbled upon this and I was wondering if anyone here could come up with a simple proof:
Let $p$ be a prime such that $p\equiv 1 \bmod 8$, and let $a,b\geq 1$ such that
$$p=a^2-2b^2.$$
...
2
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1answer
119 views
bound of quadratic nonresidue modulo an odd prime
I heard that if $p$ is an odd prime, there is a quadratic nonresidue modulo $p$ less than $p^{1/2} +1$. How to show this statement?
3
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2answers
211 views
Show that a prime divisor to $x^4-x^2+1$ has to satisfy $p=1 \pmod{12}$
Suppose that $x$ solves $x^4-x^2+1= 0 \pmod p$. Show that $p=1 \pmod {12}$. Following a hint I have rewritten the equation as $(x^2-1)^2=-3 \mod p$ and $(2x^2-1)^2=-x^2 \pmod p$. The first equation ...
0
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1answer
158 views
Understanding the set of least residues mod p, p a prime
I would like clarification\intuition about the set of least residues mod p, and the integer $\mu$.
Definition: Consider S= {$ -(p-1)/2, -(p-3)/2,\cdots ,-1,1,2,\cdots ,(p-1)2 $}.This is called ...
4
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2answers
243 views
quadratic reciprocity
happy new year
I have this statement:
"By quadratic reciprocity there are the integers $a$ and $b$ such that $(a,b)=1$, $(a-1,b)=2$, and all prime $p$ with $p\equiv a$ (mod $b$) splits in $K$ (where ...
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1answer
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Quadratic Reciprocity - Special cases
The general theorem is: for all odd, distinct primes $p, q$, the following holds:
$$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$
I've discovered the ...
6
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3answers
199 views
How can one deduce quadratic reciprocity from Hilbert reciprocity?
Hilbert reciprocity says the following:
Define $(a,b)_p$ to be $1$ if there is a non-trivial solution in $\mathbb{Q}_p$ to $z^2=ax^2+by^2$, and $-1$ if there isn't. Then $\prod_p (a,b)_p =1$, where ...
6
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2answers
285 views
Conceptual Proof of Quadratic Reciprocity
What is a conceptual proof of quadratic reciprocity? I saw the wiki proof using algebraic machinery. I don't know what field extensions, Galois groups etc are. But I want to understand how properties ...
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3answers
437 views
Explicit formula for Fermat's 4k+1 theorem
Let $p$ be a prime number of the form $4k+1$. Fermat's theorem asserts that $p$ is a sum of two squares, $p=x^2+y^2$.
There are different proofs of this statement (descent, Gaussian integers,...). ...
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1answer
199 views
Is quadratic reciprocity problem in coNP?
Quadratic reciprocity is in $\mathsf{NP}$, since to prove $x$ is quadratic residue you can show $y$ such that $y^2=x$.
Wikipedia claims the problem is in $\mathsf{coNP}$. This book claims it is not ...
5
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3answers
293 views
Roots of $x^2 + 2x + 2$
I'm trying to show that there are infinitely many values of $p$ such that $x^2 + 2x + 2$ has no roots over $\mathbb{F}_p$. Is this easily solvable? (I kind of came up with it myself so I don't ...
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1answer
550 views
How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?
My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity.
Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
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1answer
148 views
How to discover how to evaluate the Gauss sum?
Define the Gauss sum to be $$g_p(a) = \sum_{n=0}^{p-1}\left(\frac{n}{p}\right)\zeta_p^{an}.$$
We want to compute $g_p(1)^2$ (the answer is it equals $\pm p$ sign depending $p \pmod 4$). To do this ...
5
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1answer
227 views
Help complete a proof of Dirichlet on biquadratic character of 2?
I am stuck proving the theorem that there exists $x$, $x^4 \equiv 2 \pmod p$ iff $p$ is of the form $A^2 + 64B^2$.
So far I have got this (and I am not sure if it's correct)
Let $p = a^2 + b^2$ be ...
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2answers
405 views
Where does Quadratic Reciprocity point?
In his book Lectures on the theory of algebraic numbers, Hecke says that the content of the quadratic reciprocity theorem, formulated and proved entirely in terms of rationals (integers) points beyond ...
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3answers
391 views
Quadratic Congruence and Sum of Two Squares
How would one go about showing how many solutions the following congruence has?
$$x^2 + y^2 \equiv 23 \pmod{93}.$$
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7answers
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Uses of quadratic reciprocity theorem
I want to motivate the quadratic reciprocity theorem, which at first glance does not look too important to justify it being one of Gauss' favorites. So far I can think of two uses that are basic ...
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3answers
351 views
A particular case of the quadratic reciprocity law
To motivate my question, recall the following well-known fact: Suppose that $p\equiv 1\pmod 4$ is a prime number. Then the equation $x^2\equiv -1\pmod p$ has a solution.
One can show this as follows: ...
