# Questions tagged [legendre-symbol]

For questions involving the Legendre symbol, $\genfrac{(}{)}{}{}{a}{p}$ for integer $a$ and prime $p$.

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128 views

### Connection between sgn character and the Legendre symbol

Today, while I was lecturing on the Legendre symbol, I realized that the phenomenon: "the product of two non-squares is a square" isn't so foreign. For example, for $\mathbb R^\times$, the squares are ...
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### Application of Legendre, Jacobi and Kronecker Symbols

Legendre, Jacobi and Kronecker Symbols are powerful multiplicative functions in ...
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### The Number of involutory matrices over $\mathbb{Z_p}$

I want to prove the number of 2-by-2 Involutory Matrices ($A^2=I$) over $\mathbb{Z_p}$ using quadratic residue and legendre symbol. I already know that the formula is $p^2$ for characteristic of a ...
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### Analogue of Fermat’s Little Theorem

The question is “Establish an analogue of Fermat’s Little Theorem for the ring $\mathbb{Z} [\sqrt{-2}]$.” I know how to do this for the cases where $\mathbb{Z} [\sqrt{3}]$ and $\mathbb{Z} [\sqrt{5}]$...
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### Why the expansion of functions of 2 variables in legendre polynomials does not take into account all products Pi (x) Pj (y)?

My question arises from this reading link (In this article, in equation 6 on page 3, also expands a function of 2 variables in this way Parametric estimate of intensity inhomogeneities applied to MRI),...
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### Legendre symbols as multiplicative homomorphisms in number fields

$\newcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}$ Suppose we have a finite set of rational primes $B=\{p_1,\ldots,p_k\}$, and $V=\{ x\in\mathbb{Q}^*~|~x\text{ contains only primes in B} \}$. So ...
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### Legendre symbol multiplicativity

Is there some formal proof of Legendre symbol's being multiplicative? It seems pretty intuitive but I can't think of a way to formally show it?
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### Existence of a non - square solution to a modular equation

Let $p_1, \dots, p_l \in \mathbb{Z}$ be pairwise disjoint odd primes. By the Chinese remainder follows there exists $x \in \mathbb{Z}$ satisfying \begin{align*} x & = 1 \text{ mod } 4 \cdot p_2 \...
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### The sum $\sum_{r=1}^{p-1} r(r|p)$ when $p$ is an odd prime of the form $4k+3$, $k\geq 1$.

In the book Apostol Analytic Number Theory, $(r|p)$ denotes the Legendre Symbol. The exercise tell us to prove when $p\equiv 1\pmod 4$, $$\sum_{r=1}^{p-1}r(r|p)=0.$$ I can solve this quickly, but I ...
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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 ...
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### How to guarantee non existence of order 4 elements in class group of a maximal order

If we know the prime factorisation of the fundamental negative discriminant $\Delta_K$, say $n$ prime factors, then we are guaranteed that $2^{n-1}\mid h_K$, the class number of $C(\mathcal{O}_K)$ ...
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### Let $p$ be an odd prime. Suppose $a$ and $b$ are both primitive roots mod $p$. Show that $ab$ is not a primitive root mod $p$

Let $p$ be an odd prime. Suppose $a$ and $b$ are both primitive roots mod $p$. Show that $ab$ is not a primitive root mod $p$. Would appreciate some proof-checking here. First, we show that a ...
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### Legendre calculation passing through Jaccobi

We know that Jacobi's symbol does not guaranty that if $(\frac{a}{b}) = 1$ then $a$ is a quadric remander mod $b$. We also know that when calculating Legendre symbol you can use quadratic reciprocity ...
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### Number theory in the quadratic field with golden section unit

I would like to ask a favor. Does anyone of you here have an access to the book 'Number theory in the quadratic field with golden section unit' by Fred Wayne Dodd? I just need to see Theorem 8.5 of ...
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### Evaluate the Legendre Symbols (503/773)

Evaluate the Legendre Symbols (503/773) Solution: (503/773) = (270/503) = (2/503)(3^3/503)(5/503) = 1*(5/503)(3/503) = (503/5)(-1)(503/3) = -(3/5)(2/3) = -1 I don't understand how they obtain (-(3/5)...
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### Legendre Symbol by Fermat's little theorem.

My work: How do continue with part $b$, I'm so confused.
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### Showing $\sum_{b=0}^{N-1}\left(\frac{b}{p}\right)\zeta_M^{-kb} = 0$

I want to evaluate $$\sum_{b=0}^{N-1}\left(\frac{b}{p}\right)\zeta_M^{-kb}$$ for $p$ an odd prime, $p|N$, $M|N$, $(k,M)=1$. $\left(\frac{b}{p}\right)$ is the Legendre symbol. I'm fairly sure it should ...
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### The map $\left(\frac{.}{p}\right):=\mathbb F_p \rightarrow [\pm1]$ taking $a\in \mathbb F_p$ to $\left(\frac{a}{p}\right)$ is homomorphism

Prove that the map $\left(\frac{.}{p}\right):=\mathbb F_p \rightarrow [\pm1]$ taking $a\in \mathbb F_p$ to $\left(\frac{a}{p}\right)$ is a homomorphism I don't know how to even start this problem. ...
I am trying to prove: If $m,n$ are odd coprime positive integers, then $$\Big(\frac mn\Big)\Big(\frac nm\Big)=(-1)^{\large\frac{m-1}2\frac{n-1}2},$$ where $\big(\frac mn\big)$ is the Jacobi ...
Compute $(\frac{92}{11}$). Now $92^2 \equiv x^2\ mod(11)$ Now since there is no such x that satisfies this, so the legendre symbol is $-1$. Is this right? Also, can somebody explain in a slightly ...