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2
votes
1answer
203 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 ...
6
votes
1answer
203 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^{...
4
votes
1answer
198 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
0answers
126 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
0answers
133 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
1answer
382 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 $A^-_p$ ...
11
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0answers
251 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&...
13
votes
3answers
685 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 ...
14
votes
2answers
300 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
0answers
108 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
0answers
216 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
0answers
208 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
2answers
163 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 ...
6
votes
0answers
181 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 ...
0
votes
1answer
1k 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
2answers
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 ...
2
votes
1answer
257 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
2answers
252 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
3answers
352 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
2answers
242 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 ...
17
votes
4answers
3k 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
1answer
518 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
2answers
613 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
2answers
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
0answers
287 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
3answers
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 $...
10
votes
4answers
2k views

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 $\Sigma_i a^2_i$...
2
votes
1answer
517 views

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 ...
2
votes
1answer
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 ...
9
votes
2answers
680 views

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 ...
7
votes
3answers
1k views

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 $...
8
votes
3answers
848 views

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 ...
15
votes
3answers
1k views

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 $g \in {\mathbb Z}[x]$ (...