All Questions
14 questions
40
votes
1
answer
2k
views
Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
- 79.9k
13
votes
1
answer
524
views
The number of representations of an integer as the inner product of integral lattice points
I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a question I asked about four years ago. In particular I wrote that for an ...
- 2,506
6
votes
2
answers
743
views
Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer.
While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...
- 319
5
votes
0
answers
217
views
Isomorphism classes of lattices
Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define
$$
V = \{x \in \mathbb R^6 \mid A \cdot x = 0\}
$$
and
$$
\Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...
- 1,362
5
votes
0
answers
606
views
Necessary and Sufficient condition for Sharpness of Bombieri and Vaaler's result on Siegel's lemma?
This Wikipedia page currently quotes Bombieri and Vaaler's result on Siegel's lemma:
Suppose we are given a system of $m$ linear equations in $n$ unknowns such that $n>m$, say
$a_{11}x_1+\dots+...
- 13.9k
4
votes
0
answers
145
views
Number of nontrivial integral solutions to $f(x)=f(y)$
Let $f(x)\in\mathbb Z[x]$ be a nonconstant polynomial, and let $$g(x,y)=\frac{f(x)-f(y)}{x-y}\in\mathbb Z[x,y].$$ Let $N(B)$ denote the number of pairs of integers $(x_0,y_0)$ such that $1\le x_0,y_0\...
- 1,021
3
votes
1
answer
397
views
Rational points of bounded height on a variety
I would like to ask for some clarification on the following argument which I can not quite understand.
There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
- 8,998
2
votes
1
answer
269
views
System of two linear Diophantine equations
Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system
$$
\left\lbrace\begin{array}{l}
\sum_{i=1}^nx_i = 3n; \\
\sum_{i=1}^n (2i-1)x_i = ...
- 8,998
2
votes
0
answers
147
views
Genus Zero Diophantine Equations and Infinite Valuations
I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:
[1] Solving genus zero diophantine ...
- 211
1
vote
2
answers
127
views
Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...
- 3,625
1
vote
1
answer
198
views
Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application
Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil<
r_1,r_2<\lceil1+\sqrt{p}\rceil$ and
$$r_1\equiv mac\bmod p$$
$$r_2\equiv mbd\bmod p$$...
- 13.9k
1
vote
0
answers
115
views
Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
- 403
0
votes
1
answer
152
views
Almost Pell type equation
Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
- 8,998
0
votes
0
answers
147
views
Chinese remaindering to solve solvable diophantine equations
Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...
- 1,826