All Questions
30 questions
1
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0
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195
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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
$$\det\left[\left(\frac{j+k}p\right)\right]_{1\le j,k\le(p-1)/2}\ \ \text{...
- 15.6k
4
votes
0
answers
238
views
Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 15.6k
11
votes
2
answers
615
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 $...
- 17.9k
5
votes
0
answers
541
views
Two conjectures for primes $p\equiv 1\pmod 8$
Motivated by my paper Quadratic residues and quartic residues modulo primes [Int. J. Number Theory 16 (2020), 1833-1858], here I pose two new conjectures for primes $p\equiv1\pmod8$ based on my ...
- 15.6k
11
votes
0
answers
458
views
effective and unconditional upper bound for the smallest quadratic residue
Let $p$ be a prime number, and let $r=r(p)$ be the smallest prime number with $(r/p)=1$. The classical result of Linnik-Vinogradov (based on Burgess) implies that $r\ll_\epsilon p^{1/4+\epsilon}$, but ...
- 1,294
4
votes
1
answer
256
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Counting squares modulo $p$ that are also prime in an interval
What would be the best lower bound for the number of squares modulo $p$ in an interval $[1,N]$ with $N<p$ that are prime?
Via the Burgess bound, I can find a lower bound for the number of squares ...
- 41
3
votes
1
answer
427
views
Does each prime $p>541$ have a quadratic residue $x^4+y^4<p$?
For any prime $p>5$, one of the numbers
$$1^2+1=2,\ \ 2^2+1=5,\ \ 3^2+1=10=2\times5$$
is a quadratic residue modulo $p$. In 2014 I conjectured that each prime $p$ has a primitive root $g<p$ of ...
- 15.6k
2
votes
0
answers
84
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 ...
- 141
3
votes
0
answers
125
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
$$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...
- 15.6k
4
votes
0
answers
178
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Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers
The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by
$$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{...
- 15.6k
5
votes
1
answer
208
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 ...
- 841
9
votes
1
answer
970
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
$$\prod_{k=1}^{(p-1)/2}(\zeta-e^{2\pi ik^2/...
- 15.6k
4
votes
0
answers
238
views
On the values of $\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p})$ for primes $p>3$
In a recent preprint, I investigated
$$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$
where $p$ is an odd prime and $x$ is a root of unity.
Motivated by Question 337879 and Question 338325, ...
- 15.6k
4
votes
2
answers
709
views
On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity
Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
- 15.6k
9
votes
1
answer
486
views
A conjectural formula for the class number of the field $\mathbb Q(\sqrt{-p})$ with $p\equiv3\pmod8$
Question. Is my following conjecture new? How to prove it?
Conjecture. Let $p>3$ be a prime with $p\equiv3\pmod 8$, and let $h(-p)$ denote the class number of the imaginary quadratic field $\...
- 15.6k
6
votes
1
answer
367
views
Is $|\{(j,k):\ 1\le j<k\le\frac{p-1}2:\ \&\ (j^{16}\ \text{mod}\ p)>(k^{16}\ \text{mod}\ p)\}|$ even for each prime $p\equiv1\pmod {16}$?
In my paper http://arxiv.org/abs/1809.07766, I determined the parity of
$$\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|$$
for any ...
- 15.6k
1
vote
0
answers
477
views
Legendre Symbol of a Very, Very Large Value
I'm trying to use FLINT (Fast Library for Number Theory) to calculate the Legendre Symbol of the following:
$$\left(\frac{n! + 1}{p}\right)$$
In my case, $p$ is a positive, odd prime (specifically $...
- 119
6
votes
0
answers
206
views
Primes $p\in(n,2n)$ with $(\frac{-n}p)=-1$
Bertrand's postulate proved by Chebyshev states that for any $x>1$ there is a prime $p$ in the interval $(x,2x)$. In 2012 I considered some refinements of this by imposing additional requirement ...
- 15.6k
7
votes
2
answers
1k
views
A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?
I have the following conjecture involving a possible new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$ with $p$ an odd prime.
Conjecture. Let $p$ be an odd ...
- 15.6k
3
votes
0
answers
121
views
Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?
Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...
- 15.6k
2
votes
1
answer
230
views
On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$
Let $p$ be an odd prime. For $d\in\mathbb Z$ we define
$$T(d,p):=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{0\le i,j\le(p-1)/2},$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
By (1.17) of my ...
- 15.6k
4
votes
1
answer
484
views
A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (III)
As in Question 319254, for an odd prime $p$ and integers $c,d$ we define
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right),$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
In my ...
- 15.6k
1
vote
0
answers
119
views
A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (II)
As in Question 319254, for an odd prime $p$ and integers $c,d$ we let
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right).$$ If $p\equiv1\pmod4$, then we obviously have
\begin{align}&\...
- 15.6k
1
vote
0
answers
174
views
A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (I)
Let $p$ be an odd prime. Here I introduce the sum
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right)$$
with $c,d\in\mathbb Z$, where $(\frac{\cdot}p)$ is the Legendre symbol.
I have a ...
- 15.6k
2
votes
1
answer
235
views
Does $\det[\lfloor(i^2+j^2)/p\rfloor]_{1\le i,j\le(p-1)/2}$ vanish for each prime $p>7$ with $p\equiv3\pmod4$?
Let $\lfloor x\rfloor$ be the floor function.
QUESTION: Does the determinant
$$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 ...
- 15.6k
5
votes
1
answer
471
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, \...
- 397
1
vote
0
answers
223
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 $p$?...
- 15.6k
1
vote
2
answers
343
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 ...
- 51
4
votes
2
answers
297
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 ...
- 704
9
votes
3
answers
680
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} \...
- 483