Questions tagged [quadratic-residues]
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93 questions
10
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An identity related to sum of two squares
Motivation: Let $k$ be a positive integer and $m=4k+1$. I want to find the necessary conditions for the following identity to hold：
$$\displaystyle\sum_{i=1}^{k}\lfloor\sqrt{im}\rfloor=\frac{m^2-1}{12}...
- 990
2
votes
0
answers
172
views
Curious congruences involving Domb numbers
The Domb numbers given by
$$D_n:=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ (n=0,1,2,\ldots)$$
arising from enumerative combinatorics have many interesting properties (cf. https://...
- 18.1k
2
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0
answers
68
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A function through the Jacobi symbols with additional properties
For $n$ a positive odd integer and $r$ an integer, denote the Jacobi symbol by $\bigl( \frac{r}{n} \bigr)$. Eisenstein constructed a function $f(r,n)$ (with $r$ real) through the Jacobi symbols, ...
- 603
6
votes
2
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380
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Elementary question about a biquadratic Diophantine equation
Question: For an odd prime $p$, if there exist integers $x,y,z$ such that $x^4-4y^4=pz^2$ and $x,z\equiv1 \text{(mod 2)}$, is it true that $p\equiv1 \text{(mod 16)}$ ?
If we suppose that $x,y,z$ are ...
- 990
7
votes
0
answers
422
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Average of $\left(\frac{p}{q}\right)$ over $p\leq q^\epsilon$ for *most* $q$? (unconditionally)
Let $(\cdot/\cdot)$ be the Jacobi symbol. Consider the problem of showing that $(p/q)$ averages to $0$ as $p$ ranges over the primes $\leq q^\epsilon$, or some smaller range. For fixed $q$ (prime or ...
- 22k
11
votes
1
answer
920
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A conjecture on quadratic residues
Given $p$ is an odd prime, let $a_1, a_2,\dots, a_p$ be integers such that $p\mid (a_1+a_2+\dots+a_p)$ and that ,
$$
\frac{a_{j+1}+a_{j+2}+\dots+a_{j+i}}{i}\quad \text{is a quadratic residue of $p$ (...
- 121
1
vote
1
answer
217
views
Estimates for number of primes satisfying certain quadratic residuosity condition
Given a square-free integer $d$, how many primes $p<d$ such that $p$ is a quadratic residue modulo $d$ and $\left(\frac{d}{p}\right)=1$?
Also, what are the estimates for the same problem with ...
- 387
1
vote
0
answers
225
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Conjectural values of some determinants involving Legendre symbols (II)
Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by the evaluation of the determinants
$$\det\left[\left(\frac{j+k}p\right)\right]_{1\le j,k\le(p-1)/2}\ \ \text{...
- 18.1k
4
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Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 18.1k
0
votes
0
answers
81
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Factorial quadratic residue
I would like to find all positive integers $n$ and $m$ such that $n^2 \equiv m! \ ( \text{mod } 2024)$. I see that for $m=1$ there is $n=45$ such that the relation holds.
I think that there is no ...
11
votes
2
answers
809
views
Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
- 18.5k
8
votes
4
answers
1k
views
A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
- 8,665
0
votes
1
answer
208
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On a summation in "Artin's conjecture for primitive roots" by Heath-Brown
This is a question on the paper: D. R. Heath-Brown. Artin's conjecture for primitive roots. Quarterly J. Math. 37 (1): 27–38, 1986.
At the beginning of the proof of his main theorem on page 35, Heath-...
- 3
2
votes
2
answers
385
views
Expressions for binomial residue sum $\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$
I'm interested in the sum:
$$\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$$
where $q$ is a prime number. This is just the binomial expansion with an extra weight on quadratic residues ...
- 591
1
vote
0
answers
114
views
Functions that take quadratic residues to non quadratic residues
Let $p$ be prime and $Q$ be the set of integers $x$ mod $p$ so that $x^2-1$ is a quadratic residue. Let $Q^c$ be the complement of $Q$. If we don't consider $x = 1$ then these two sets have the same ...
- 591