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2answers
215 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
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1answer
236 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 $a(1),a(2),a(3),\dots$ ...
4
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2answers
196 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 ...
5
votes
3answers
254 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} ...
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2answers
218 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
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4answers
1k 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
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1answer
396 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
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2answers
541 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
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2answers
936 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
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0answers
222 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
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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 ...
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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
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1answer
425 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
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1answer
339 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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0answers
730 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
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2answers
527 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 ...
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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
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3answers
772 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 ...
14
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3answers
824 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 $g \in {\mathbb Z}[x]$ ...