# Questions tagged [quadratic-residues]

The quadratic-residues tag has no usage guidance.

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### Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?

Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...

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152 views

### On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$

Let $p$ be an odd prime. For $d\in\mathbb Z$ we define
$$T(d,p):=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{0\le i,j\le(p-1)/2},$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
By (1.17) of my ...

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### Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by
$$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$
In 2004, R. Chapman [Acta ...

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### On the determinant $\det[(i^2+dj^2)(\frac{i^2+dj^2}p)]_{1\le i,j\le(p-1)/2}$ with $p$ an odd prime

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. We define the determinant $D(d,p)$ by
$$D(d,p):=\det\left[(i^2+dj^2)\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}...

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### On the sum $\sum_{x=0}^{(p-1)/2}(\frac{x^{4n}+cx^{2n}+d}p) $ with $p$ an odd prime

Let $p$ be an odd prime, and let $n$ be a positive integer. For $c,d\in\mathbb Z$ we define
$$F_p^{(n)}(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^{4n}+cx^{2n}+d}p\right),$$
where $(\frac{\cdot}p)$ is ...

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313 views

### A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (III)

As in Question 319254, for an odd prime $p$ and integers $c,d$ we define
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right),$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
In my ...

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### A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (II)

As in Question 319254, for an odd prime $p$ and integers $c,d$ we let
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right).$$ If $p\equiv1\pmod4$, then we obviously have
\begin{align}&\...

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### A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (I)

Let $p$ be an odd prime. Here I introduce the sum
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right)$$
with $c,d\in\mathbb Z$, where $(\frac{\cdot}p)$ is the Legendre symbol.
I have a ...

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198 views

### Permutations of squares and finite fields

Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let
$$S(n)=\bigg\{\sum_{k=1}^nk^2\pi(k)^2:\ \pi\in S_n\}.$$
Motivated by Question 316142 of mine, here I ask the following ...

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163 views

### Does $\det[\lfloor(i^2+j^2)/p\rfloor]_{1\le i,j\le(p-1)/2}$ vanish for each prime $p>7$ with $p\equiv3\pmod4$?

Let $\lfloor x\rfloor$ be the floor function.
QUESTION: Does the determinant
$$D_p=\det\left[\left\lfloor\frac{i^2+j^2}p\right\rfloor\right]_{1\le i,j\le(p-1)/2}$$ vanish for each prime $p>7$ with ...

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219 views

### On triangular numbers modulo primes

Let $p$ be an odd prime. For $a\in\mathbb Z$ let $\{a\}_p$ denote the least nonnegative residue of $a$ modulo $p$. The list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ is a permutation of all the quadratic ...

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213 views

### On $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i+j}(2i+j)$ and $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i-j}(2i-j)$ modulo a prime $p>3$

QUESTION: Is my following conjecture true?
Conjecture. Let $p>3$ be a prime and let $h(-p)$ be the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. Then
$$\frac{p-1}2!!\prod^{...

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209 views

### Quadratic Nonresidue

Suppose $r$ integers $n_1, \ldots, n_r$ are given such that $0<|n_i|<N$ for $1 \leq i \leq r$ and for a natural number $N.$ Is it possible to find a prime number $p$ such that all numbers $n_1, \...

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### Congruence for the product of quadratic residues + the product of quadratic non-residues

My question has been here on MSE for a long time, but it has not received a full answer. I bring it here:
Find a prime $p$ such that $p \equiv 1 \bmod 4$ and such that the
product in the range $[...

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### Evaluate a curious determinant with Legendre symbol entries

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. R. Chapman's conjecture on the exact value of the determinant of
$$C_p:=\left[\left(\frac{i-j}p\right)\right]_{0\le i,j\le (p-...

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395 views

### A new determinant question for primes $p\equiv3\pmod4$

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by my question http://mathoverflow.net/questions/310301, here I introduce the matrices $A^+_p$ and $A^-_p$ ...

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271 views

### A determinant problem for primes $p\equiv 1\pmod4$

Let $p$ be an odd prime, and let $A_p$ denote the matrix
$$[a_{ij}]_{1\le i,j\le (p-1)/2},$$
where
$$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...

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### Is $-\det\big[\big(\frac{i^2+j^2}p\big)\big]_{1\le i,j\le (p-1)/2}$ always a square for each prime $p\equiv 3\pmod 4$?

Let $p$ be an odd prime and let $S_p$ denote the determinant
$$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$
with $(\frac{\cdot}p)$ the Legendre symbol. By Theorem 1.2 of my ...

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### Does $n^2$ divide $\det\left[\left(\frac{i^2+2ij+3j^2}n\right)\right]_{1\le i,j\le n-1}$ for each odd integer $n>3$?

For any odd integer $n>1$ and integers $c$ and $d$, define
$$(c,d)_n:=\det\left[\left(\frac{i^2+cij+dj^2}n\right)\right]_{1\le i,j\le n-1},$$
where $(\frac{\cdot}n)$ is the Jacobi symbol. It is ...

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### Does each prime $p>3$ have a quadratic nonresidue which is a Mersenne number?

Recall that the Mersenne numbers are those integers $M_p=2^p-1$ with $p$ prime.
QUESTION: Is it true that for each prime $p>3$ there is a Mersenne number which is a quadratic nonresidue modulo $p$?...

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### Expressing quartic Dirichlet characters modulo primes $p\equiv 1\bmod 4$ with Legendre symbols

Looking through some old notes of mine from two years ago I found some crude notes writing what amounted to the statement that for any prime $p\equiv 1\bmod 4$ one could express for any odd integer $p\...

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### some phenomena about Fibonacci number [closed]

Edit: for conjecture 3, define :$(\frac{5}{F_n}) = 5 ^{\frac{F_n-1}2} mod (F_n)$, $F_n $ is odd number.
When study Fibonacci number, some phenomena are observed：
$(\frac{F_p}{5})= (\frac{p}{5}) ...

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### Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?

Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?
Using the Tonelli-Shanks algorithm and the Chinese remainder ...

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### Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the
first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues".
I recently ...

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### Relationship between quadratic residues modulo a prime and quadratic residues modulo a prime power [closed]

Been going through Alan Baker's A Comprehensive Course in Number Theory. Very interesting book, although the way proofs are presented sometimes throws me off a little.
I usually read through a ...

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287 views

### overlap quadratic residues

Let $p$ be a prime number of form $4k+1$ and $M$ is its quadratic residue set.
Let $M_i=\{i+x|\forall x\in M\}$ $\forall 0<i<p$.
Does there exist a positive constant $\varepsilon$ such that ...

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### How does this sequence grow

Let $a(n)$ be the number of solutions of the equation $a^2+b^2\equiv -1 \pmod {p_n}$, where $p_n$ is the n-th prime and $0\le a \le b \le \frac{p_n-1}2$. Is the sequence $a(1),a(2),a(3),\dots$ non-...

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### Orders of the conjugates of an algebraic prime number in its residue field

Of interest to me is the following question (it would be nice to find out what is known in its direction):
Given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime ...

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### Quadratic residues and nonresidues of arbitrary patterns

Let $p_1, p_2, \dotsc, p_n$ be distinct primes, and let $\epsilon_1, \epsilon_2, \dotsc, \epsilon_n$ be an arbitrary sequence of $1$ and $-1$.
There is an integer $a$ such that $\left( \frac{a}{p_1} \...

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243 views

### Impossible Range for Minkowski-Like Sum of Squares

Given coprime positive integers M,N, and a corresponding integer z outside of the range (for all integers x,a,b,c) of $Mx^2-N(a^2+b^2+c^2)$, is there any such z which is "deceptive", meaning that it ...

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### Does there exist a non-square number which is the quadratic residue of every prime?

I want to know whether there exist a non-square number $n$ which is the quadratic residue of every prime.
I know it is very elementary, and I think those kind of number are not exist, but I don't know
...

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### Are there Carlitz analogues of quadratic residues and reciprocity?

Let $q$ be a prime power. I will use the notations of Keith Conrad's Carlitz extensions paper (but I'll work over $\mathbb{F}_q$ rather than $\mathbb{F}_p$).
The most general question I'm asking here ...

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### quadratic residues - is there an easy explanation for the pattern I'm seeing?

Let $m$ be an integer and $q$ be an odd prime factor of $m^2 + 1$. Is there an obvious reason that $\left(\frac{2m}{q}\right)$ always equals 1? From some numerics, this seems to be the case.
The ...

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### Three consecutive quadratic residues problem

Prove that doesn't exist $N\in\mathbb{N}$ with property: for all primes $p>N$ exist $n\in\{3, 4,\ldots, N\}$ such that $n, n-1, n-2$ are quadratic residues modulo $p$.

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### Shortest interval over which there are more quadratic residues than nonresidues

Hi, I refer to formula (8) in Chapter 1 of H. Davenport, Multiplicative Number Theory, Third Edition, Springer (2000), which says that for primes $q\equiv 3 \bmod 4$:
$$
L\left(\left(\frac{\cdot}{q}\...

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### Gauss sum (with sign) through algebra

Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$.
Let $G = \sum\limits_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the standard Gauss sum for $...

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### Sum of squares modulo a prime

What is the probability that the sum of squares of n randomly chosen numbers from $Z_p$ is a quadratic residue mod p?
That is, let $a_1$,..$a_n$ be chosen at random. Then how often is $\Sigma_i a^2_i$...

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### Numbers with few quadratic residues

It is well known that the upper bound on the number of quadratic residues mod n is approximately n/2 and it reaches this bound for n prime.
Is there any similar lower bound on the number of quadratic ...

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364 views

### Distribution of quadratic residues of a fixed number without using Dedekind zeta function

Let $n > 1$ be a square-free natural number, which is fixed. The assertion to be proved is the following:
Let $p$ run through primes. Then, $$\left( \frac{n}{p} \right)$$ is equally distributed ...

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### Positivity of $L(1,\chi)$ for real Dirichlet's character

Let $\chi$ be a real nonprincipal Dirichlet's character modulo $m$.
In my
answer to the question on $L(1,\chi)$, I explain a trick for showing that $L(1,\chi)>0$ on the simplest examples
of the ...

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### Intuition for a formula that expresses the class number of an imaginary quadratic field by counting quadratic residues

If $p$ is a prime of the form $4n+3$, the class number $h$ of $Q[\sqrt{-p}]$ can be expressed using the number $V$ of quadratic residues and $N$ nonresidues in the interval $[1,\frac{p-1}{2}]$:
If $...

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### Isolated quadratic residues in integers mod p

For prime p sufficiently large, there is always an integer q such that q is a residue mod p, but neither q−1 nor q+1 are; the number of such residues scales like p/8 (and similarly for any ...

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### Irreducibility of polynomials related to quadratic residues

Let $p \equiv 1 \bmod 4$ be a prime number. Define the polynomial
$$ f(x) = \sum_{a=1}^{p-1} \Big(\frac{a}{p}\Big) x^a. $$
Then $f(x) = x(1-x)^2(1+x)g(x)$ for some polynomial
$g \in {\mathbb Z}[x]$ (...