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5
votes
3answers
238 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
214 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 ...
9
votes
4answers
900 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
355 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
531 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 ...
6
votes
2answers
909 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
209 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 $q\equiv 3 \bmod 4$: $$ ...
8
votes
3answers
1k 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
1k 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 ...
2
votes
1answer
415 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
337 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 ...
0
votes
0answers
706 views

Efficient quadratic residue mod 2^32

I want to determine if a value is a quadratic residue mod $2^{32}$. I've developed a very fast pre-screening method based on a Bloom Filter that identifies quadratic residues for mod $2^7=128$ in ...
9
votes
2answers
519 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
956 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
759 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 ...
13
votes
3answers
794 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 $g \in {\mathbb Z}[x]$ ...