# Questions tagged [quadratic-residues]

The quadratic-residues tag has no usage guidance.

57
questions

**9**

votes

**3**answers

506 views

### Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

As the title asks: does there exist $N$ such that, for any prime $p$ larger than $N$, the expression $x^4 +y^4$ takes on all values in $\mathbb{Z}/p\mathbb{Z}$?
I have been thinking about this ...

**2**

votes

**0**answers

62 views

### quadratic residues and cubic polynomials [closed]

I'm really not sure about this, but I've heard somewhere that for any prime $p$,
$|\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} ) |\le \sqrt{2p}$ holds.
Does anyone know a proof for this inequality ...

**2**

votes

**0**answers

102 views

### On the set $\{n>0:\ n\ \text{is a quadratic nonresidue modulo the}\ n\text{th prime}\}$

Let $S$ denote the set of positive integers $n$ with $n$ a quadratic nonresidue modulo the $n$th prime $p_n$. The first 20 elements of $S$ are
$$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...

**4**

votes

**0**answers

86 views

### Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers

The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by
$$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$
and
$$L_0=2,\ L_1=1,\ \text{...

**5**

votes

**1**answer

169 views

### applications of finding least quadratic nonresidue mod $p$?

I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $p$.
My question is that why it is so ...

**9**

votes

**1**answer

855 views

### A conjecture on primitive tenth roots of unity

QUESTION. How to solve my following conjecture involving primitive tenth roots of unity?
Conjecture. Let $\zeta$ be any primitive tenth root of unity. Then
$$\prod_{k=1}^{(p-1)/2}(\zeta-e^{2\pi ik^2/...

**4**

votes

**0**answers

199 views

### On the values of $\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p})$ for primes $p>3$

In a recent preprint, I investigated
$$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$
where $p$ is an odd prime and $x$ is a root of unity.
Motivated by Question 337879 and Question 338325, ...

**2**

votes

**2**answers

553 views

### On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity

Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...

**9**

votes

**1**answer

354 views

### A conjectural formula for the class number of the field $\mathbb Q(\sqrt{-p})$ with $p\equiv3\pmod8$

Question. Is my following conjecture new? How to prove it?
Conjecture. Let $p>3$ be a prime with $p\equiv3\pmod 8$, and let $h(-p)$ denote the class number of the imaginary quadratic field $\...

**6**

votes

**1**answer

298 views

### Is $|\{(j,k):\ 1\le j<k\le\frac{p-1}2:\ \&\ (j^{16}\ \text{mod}\ p)>(k^{16}\ \text{mod}\ p)\}|$ even for each prime $p\equiv1\pmod {16}$?

In my paper http://arxiv.org/abs/1809.07766, I determined the parity of
$$\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ (j^2\ \text{mod}\ p)>(k^2\ \text{mod}\ p)\right\}\right|$$
for any ...

**1**

vote

**0**answers

234 views

### Legendre Symbol of a Very, Very Large Value

I'm trying to use FLINT (Fast Library for Number Theory) to calculate the Legendre Symbol of the following:
$$\left(\frac{n! + 1}{p}\right)$$
In my case, $p$ is a positive, odd prime (specifically $...

**0**

votes

**1**answer

101 views

### Shifting quadratic residues

Let $p\equiv 3\pmod 4$ and $G$ be the set of nonzero quadratic residues modulo $p$ (so $G=(p-1)/2$). For $1\leq a\leq p-1$, let $G_a=\{(a+g)\pmod p\mid g\in G\}$. What is the size of $G_0\cap G_a$?
I ...

**6**

votes

**0**answers

186 views

### Primes $p\in(n,2n)$ with $(\frac{-n}p)=-1$

Bertrand's postulate proved by Chebyshev states that for any $x>1$ there is a prime $p$ in the interval $(x,2x)$. In 2012 I considered some refinements of this by imposing additional requirement ...

**5**

votes

**0**answers

176 views

### On the determinants $\det\left[(i\pm j)\left(\frac{i\pm j}p\right)\right]_{1\le i,j\le(p-1)/2}$

Let $p$ be an odd prime and define
$$D_p^+:=\det\left[(i+j)\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}$$
and $$D_p^{-}:=\det\left[(i-j)\left(\frac{i-j}p\right)\right]_{1\le i,j\le(p-1)/2},$$
...

**1**

vote

**0**answers

126 views

### On $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le p-1}$ and $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le (p-1)/2}$

QUESTION. Is my following conjecture true? If true, how to prove it?
Conjecture. Let $p$ be a prime with $p\equiv5\pmod 6$, and define the matrices $A_p$ and $B_p$ by
$$A_p:=\left[\frac1{i^2-ij+j^2}\...

**7**

votes

**2**answers

698 views

### A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?

I have the following conjecture involving a possible new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$ with $p$ an odd prime.
Conjecture. Let $p$ be an odd ...

**3**

votes

**0**answers

108 views

### 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 ...

**2**

votes

**1**answer

182 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 ...

**6**

votes

**0**answers

232 views

### 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 ...

**4**

votes

**1**answer

358 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 ...

**1**

vote

**0**answers

105 views

### 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}&\...

**1**

vote

**0**answers

146 views

### 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 ...

**1**

vote

**2**answers

216 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 ...

**2**

votes

**1**answer

178 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 ...

**2**

votes

**1**answer

261 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 ...

**7**

votes

**1**answer

231 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^{...

**5**

votes

**1**answer

284 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, \...

**4**

votes

**0**answers

223 views

### 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 $[...

**2**

votes

**0**answers

169 views

### 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-...

**7**

votes

**1**answer

423 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 $A^-_p$ ...

**13**

votes

**0**answers

310 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&...

**14**

votes

**3**answers

880 views

### 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 ...

**13**

votes

**2**answers

362 views

### 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 ...

**1**

vote

**0**answers

121 views

### 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 $p$?...

**7**

votes

**0**answers

246 views

### 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\...

**1**

vote

**0**answers

219 views

### 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}) ...

**3**

votes

**2**answers

268 views

### 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 ...

**7**

votes

**0**answers

212 views

### 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 ...

**1**

vote

**1**answer

2k views

### 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 ...

**1**

vote

**2**answers

302 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 ...

**2**

votes

**1**answer

261 views

### 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-...

**4**

votes

**2**answers

259 views

### 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 ...

**7**

votes

**3**answers

448 views

### 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} \...

**1**

vote

**2**answers

245 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 ...

**20**

votes

**4**answers

4k views

### 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
...

**5**

votes

**1**answer

539 views

### 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 ...

**3**

votes

**2**answers

651 views

### 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 ...

**8**

votes

**2**answers

1k views

### 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 $p$.

**4**

votes

**0**answers

304 views

### 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}\...

**11**

votes

**3**answers

2k views

### 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 $...