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Questions tagged [jacobi-symbol]

For questions on Jacobi symbols, a generalization of the Legendre symbol introduced by Jacobi in 1837.

The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography.

Source: https://en.wikipedia.org/wiki/Jacobi_symbol

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Jacobi symbol properties

I know that $p_i$ is an odd prime, $(\frac{a}{p_i}\equiv a^{\frac{p_i-1}{2}} \pmod p$). But I don't know how to solve this problem...
JooE
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Jacobi Symbol: $\sum_{n=1}^{p}\left(\sum_{m=1}^{h}\left(\frac{m+n}{p}\right)\right)^2=h(p-h)$

Show that if $p$ is and odd prime and $h$ is an integer, $1\le h \le p$, then $$\displaystyle\sum_{n=1}^{p}\left(\sum_{m=1}^{h}\left(\frac{m+n}{p}\right)\right)^2=h(p-h)$$ where $\left(\frac{m+n}{p}\right)$ denotes the Jacobi symbol. My…
user715501
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Sum of Jacobi symbol numerators

According to Theorem 2 of K. Ono, S. Robins, and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes Math. 50 (1995), 73–94., define with $\left( \frac{r}{n} \right)$ being the Jacobi symbol, $$R_3(n) =…
qwr
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$N$ square-free $\Rightarrow \exists a \in (\mathbb Z / N\mathbb Z)^{\times}, (\frac{a}{N}) = -1$

I want to show that if $N$ is a square free odd integer then there is some number coprime to $N$ such that the Jacobi Symbol $(\frac{a}{N}) = -1 $ Honestly I don't even know how to start showing this. I know that for every odd prime $p$ there are…
user366818
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Find all prime p such that Legendre symbol of $\left(\frac{10}{p}\right)$ =1

In the given question I have been able to break down $\left(\frac{10}{p}\right)$= $\left(\frac{5}{p}\right)$ $\left(\frac{2}{p}\right)$. But what needs to be done further to obtain the answer.
Yogesh Ghaturle
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Quadratic Gauss Sum with Jacobi Symbol of even nonsquare free Modulus

Let $d, \ell$ be integers, $\left(\frac{\cdot}{\cdot}\right)$ be the Jacobi symbol, I would like to compute the following Gauss…
Joseph Leung
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Generalization of Jacobi symbol for composite $n$

Suppose $n$ and $m$ are relatively prime integers. Define the symbol (sort of like the Jacobi symbol) U$(n,m)=1$ if and only if each prime $p|n$, there is an integer $k$ such that $n^k = p\pmod m$, and otherwise, U$(n,m)=-1$. This implies if $n$ is…
J. Linne
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Prove that if $a^{(n-1)/2}\equiv\pm1\pmod{n}$, then $\left(\frac{a}{n}\right)\equiv a^{(n-1)/2}\pmod{n}$

Let $a,n\ \in \mathbb Z$ and suppose that $n>1$ is odd, $n\equiv3\pmod{4}$, and that $\gcd(a,n)=1$. Prove that if $a^{(n-1)/2}\equiv\pm1\pmod{n}$, then $$\left(\frac{a}{n}\right)\equiv a^{(n-1)/2}\pmod{n}$$ I have no idea how to prove the…
Mjoseph
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Is it true that $(2k+1)(2^{4k+1}+(2k-1)^{4k+1})=a^2+(4k+1)b^2$ has no positive integer solution?

Is it true that the equation $$(2k+1)(2^{4k+1}+(2k-1)^{4k+1})=a^2+(4k+1)b^2$$ has no positive integer solution ? Let $A=(2^{4k+1}+(2k-1)^{4k+1})/(2k+1)$, if $q$ is a prime factor of $A$ such that $x^2+4k+1\equiv 0 \mod q$ has no integer solution,…
lsr314
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Understanding a Jacobi Symbol Proof

Suppose that $a,b \in \mathbb{N}$ are both odd and $m \in \mathbb{Z}.$ I want to show that $$\left(\dfrac{m}{ab}\right) = \left(\dfrac{m}{a}\right) \left(\dfrac{m}{b}\right)$$ Here, we are talking about the Jacobi Symbol, hence why I put the $\ell$…
John W. Smith
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Calculate Jacobi symbol

Calculate the Jacobi symbol $\left(\frac{n^4+n^2+1}{2n^2+1} \right)$ for every integer $n>0$. Using the properties of the Jacobi symbol, $$\left(\frac{n^4+n^2+1}{2n^2+1} \right)=\left(\frac{2n^2+1}{n^2-n+1} \right)\left(\frac{2n^2+1}{n^2+n+1}…
Dr. Heinz Doofenshmirtz
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Jacobi Symbol: $\sum_{n=1}^{n=p}\left(\frac{n^2+a}{p}\right)=-1$

Show that if $(a,p)=1$, $p$ an odd prime then, $\sum_{n=1}^{n=p}\left(\frac{n^2+a}{p}\right)=-1$, where $\left(\frac{n^2+a}{p}\right)$ is the Jacobi symbol. This question has been taken from the book : An introduction to theory of numbers by Niven,…
user715501
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Jacobi symbol:$(\frac{7m^2-1}{18m^2+1})=(\frac{25m^2}{18m^2+1})=1$?

By definition of Jacobi symbol, $(\frac{q}{n})=1$ if there exists $x$ such that $x^2 \equiv q \pmod n$ where $n$ is odd. So, $(\frac{7m^2-1}{18m^2+1})$ is equivalent to- $x^2 \equiv 7m^2-1 \pmod {18m^2+1} \implies x^2+18m^2+1 \equiv 7m^2-1+18m^2+1…
Consider Non-Trivial Cases
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Factoring out Kloosterman Sum

I'm reading Iwaniec's book and he says Kloosterman sum factors into $S(n,n;c)=S(n\bar{q},n\bar{q};r)T(n\bar{r},n\bar{r};q)$ where $n$ is square free and $c=rq$ such that $(q,n)=(q,r)=1$(i.e. $q$ is the largest factor of $c$ coprime to $n$) and $T$…
J.Shim
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Jacobi symbol flipping and coprime

The law of quadratic reciprocity for Jacobi symbols says if m and n are positive coprimes intergers then: $\left(\frac{m}{n}\right)\left(\frac{n}{m}\right) = (-1)^{\tfrac{m-1}{2}\cdot\tfrac{n-1}{2}} = \begin{cases} 1 & \text{if } n \equiv 1 \pmod…
Kurt Roeckx
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