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One of the most well known exponential diophantine equations is $(x^n-1)/(x-1)=y^p$, where $x,n,y,p$ are positive Integers with $n>2$, $x>1$, and $p$ prime. Some solutions are obtained with $x=3,...
Euro Vidal Sampaio's user avatar
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There is a rather confusing state of affairs at Wikipedia concerning Mills' constant. The article on formula for primes mentions that It is not known whether it is irrational, but the article on Mills'...
Euro Vidal Sampaio's user avatar
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Let $J$ be an abelian variety over a number field $K$, and $x \in J(K)$ be a $K$-ratinal point of $J$. Suppose that for infinitely many non-archimedean valuation $v$ of $K$, (where $J$ has good ...
Hugo's user avatar
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Is there a closed form for all integer solutions of $a^2 + b^2 + ab = c^2$ ? There are infinitely many solutions. These are the smallest positive reduced solutions (no common divisor): $(3, 5, 7), (5, ...
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Let $K$ be a number field, Galois over the rationals. Let $\alpha \in \mathbb{C}$ be an approximation to an element in $K$ which is known to have bounded height (for some sense of "height"). ...
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Let $A$ be an abelian variety over a number field $K$ of dimension $g$. Consider $\mathcal{L}$ be a symmetric ample line on $A$ that gives an embedding $ \ i_{\mathcal{L}} : A(\overline{K}) \...
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Suppose $K$ is a finitely generated field over $\mathbb Q$, then on $\bar K$ we have Moriwaki height. So I wonder if there's an analogue of Baker's theorem on finitely generated field of char $0$? I ...
Richard's user avatar
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Let $K$ be a non-archimedean local field (e.g., $\mathbb{Q}_p$ ), $A / K$ an abelian variety of dimension $g$. Consider $\mathcal{L}$ a line bundle on $A$ and $F \in \Gamma(A, \mathcal{L})$ a global ...
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I am a number theory graduate student and I've got interest in Diophantine Geometry recently. I don't have any background in Algebraic Geometry and due to which I am struggling a lot. I want to learn ...
NumDio's user avatar
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Let $F$ be a homogeneous polynomial of degree $L$ in $K[X_0, X_1,...., X_N]$ and $P$ is a point of $N$ dimensional projective space over $K$. If the logarithm Weil height of $P$ is $h(P)$ and $h(F)$ ...
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I am aware of a single variable version due to Bombieri and Vaaler that provide a non-zero polynomial of low height defined over a number field vanishing at a finite set of point with prescribed order....
NumDio's user avatar
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Let $(A, L)$ be a principally polarized Abelian variety defined over $\overline{\mathbb{Q}}$, of dimension $g$, with $L$ ample and symmetric (Line bundle correspond to the theta - $\Theta$ divisor). ...
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Let $K$ be a number field and $\mathcal{P}$ be a prime ideal of $K$. Let $k_{\mathcal{P}}$ be the residue field $\mathcal{O}_K/\mathcal{P}$. Consider the following construction used very often in ...
Breakfastisready's user avatar
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Suppose $K$ is a number field, $S$ a finite set of places of $K$, and $A$ is an abelian scheme over $\mathcal O_{K,S}$. I want to ask is there some connections between the Faltings' height $h_F(A)$ ...
Richard's user avatar
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Background: It is not known if any point exists on the XY plane that is at rational distance from all 4 vertices of a unit square lying on the XY plane (https://mathworld.wolfram.com/...
Nandakumar R's user avatar
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We add a little to Tiling the plane with pairwise non-congruent rational triangles Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "...
Nandakumar R's user avatar
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A triangle with all side lengths rational is said to be a rational triangle. It is known - for example, Cutting the unit square into pieces with rational length sides - that the unit square allows ...
Nandakumar R's user avatar
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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?
Puzzled's user avatar
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I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$. $\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \...
Breakfastisready's user avatar
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2 answers
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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 ...
Johnny T.'s user avatar
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Background on heights Consider $P = [a: b] \in \mathbb{P}^1(\mathbb{Q})$, where $a,b$ are coprime integers. We define the naive (multiplicative) height as $$H(P) = \max \{|a|, |b|\}$$ We can change ...
dummy's user avatar
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Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system $$ \left\lbrace\begin{array}{l} \sum_{i=1}^nx_i = 3n; \\ \sum_{i=1}^n (2i-1)x_i = ...
Puzzled's user avatar
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7 votes
2 answers
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Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places. It is an ...
manifold's user avatar
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Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and let $L$ be a line bundle on $A$. Then a Néron–Tate height $\hat h_L$ can be defined by taking a model $\mathcal A$ of $A$ (over the ring ...
manifold's user avatar
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Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and $$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
var's user avatar
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I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
finiteness's user avatar
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Let $a,b$ be coprime and say $0<a<b<2a$. Consider the quadratic system: $$\alpha\delta-\beta\gamma=1$$ $$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
Turbo's user avatar
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3 votes
1 answer
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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\...
Puzzled's user avatar
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6 votes
2 answers
878 views

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 ...
ASP's user avatar
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129 views

I am studying the article Around the Chevalley-Weil Theorem by Zannier, Turchet and Corvaja and I am stuck in the following point: let $V$ and $W$ be two complex quasi-projective varieties and $\pi\...
cartesio's user avatar
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Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
roydiptajit's user avatar
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3 votes
0 answers
442 views

Let $i:X\to Y$ be a closed immersion of smooth projective varieties over $\mathbb{Q}$. Assume that $Y(\mathbb{Q})$ is infinite and $X(\mathbb{Q})\to Y(\mathbb{Q})$ is surjective. Also assume that $X$ ...
guido's user avatar
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0 answers
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Let $(x_i,y_i)\in\mathbb Z^2$ at $i\in\{1,2,3,4\}$ be four (not five and I assume an unique curve exists because of Diophantine constraints and not geometric constraints) non-cocyclic integral points ...
Turbo's user avatar
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2 votes
0 answers
234 views

Suppose $q^2-4pr<0$, and consider the set of integral points $$\mathcal Z=\{(X,Y)\in\mathbb Z^2:$$ $$px^2+qxy+ry^2+sx+ty+u=0\}$$ which lie on an ellipse. Then define $$M=\sum_{(X,Y)\in\mathcal Z}\...
Turbo's user avatar
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6 votes
1 answer
274 views

Let $V/\mathbb{Q}$ be a subvariety of $\mathbb{P}^n$. There are many plausible choices of height function, some differing only by constant factors: $\max |x_i|$ (for $(x_0,x_1,\dotsc,x_n)$, $\gcd(x_1,\...
H A Helfgott's user avatar
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0 votes
0 answers
408 views

I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$. Lattice point means integer coordinates and equation with integer means diophantine ...
user2548436's user avatar
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1 vote
1 answer
229 views

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 $$r_1\equiv mac\bmod p$$ $$r_2\equiv mbd\bmod p$$...
Turbo's user avatar
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15 votes
1 answer
567 views

I have been reading about transfinite diameter and its applications to number theory and have been hunting for the following paper for quite a while: Cantor D.: On an extension of the definition of ...
asrxiiviii's user avatar
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0 answers
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I have been reading about the Schinzel-Zassenhauss conjecture, and have been looking for the following reference: Vinberg E.: On some number-theoretic conjectures of V. Arnold, Japanese J. Math., vol ...
asrxiiviii's user avatar
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0 answers
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According to page 1-2 of this paper (https://arxiv.org/abs/0906.4286), Mahler has established the inequality $$|\alpha_1 - \alpha_2| \geq H(\alpha_1)^{-(d-1)} \tag 1 $$ to be valid for all pairs of ...
asrxiiviii's user avatar
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1 vote
0 answers
75 views

If $b_{ij}\in[0,2^t-1]\cap\mathbb Z$ holds at every $i\in\{1,\dots,k'\}$ and $j\in\{1,\dots,k\}$ where $1\leq k'\leq k$ holds then there are $x_1,\dots,x_k\in\mathbb Z:\sum_{j=1}^kx_jb_{1i}=0\wedge\...
VS.'s user avatar
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0 votes
0 answers
190 views

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 ...
VS.'s user avatar
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6 votes
1 answer
604 views

The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational. To cut a unit square into n (a finite ...
Nandakumar R's user avatar
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2 votes
0 answers
213 views

In my research I am using the notion of a $\textbf{rational function f on a domain}\: U\subset \mathbb C^{n}$. By this I mean that the graph of $f$ is an analytic component of an affine variety $X$ ...
Espace' etale's user avatar
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14 votes
0 answers
552 views

Let's recall that: (1): The Ax-Kochen principle says that if $\mathcal{U}$ is a non-principal ultrafilter over prime numbers, then $\prod_{\mathbb{U}} \mathbb{F}_p((t)) \equiv \prod_{\mathbb{U}} \...
Mohammad Golshani's user avatar
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6 votes
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353 views

I am about to give a couple of lectures on Bombieri-Pila/the determinant method. Bombieri-Pila gives a bound $$|C(\mathbb{Q}) \cap B \cap \mathbb{Z}^2| \ll_{d,\epsilon} N^{1/d+\epsilon}$$ for $C$ a ...
H A Helfgott's user avatar
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In Proposition (2) in the paper [1], in below, it is proved that: Assume that $K$ is a number field and that $C$ has infinitely many points of degree $\leq d$ over $K$. (Here $d\geq 2$ is an integer)...
user131222's user avatar
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1 vote
0 answers
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Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
VS.'s user avatar
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1 vote
0 answers
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Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
Turbo's user avatar
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5 votes
0 answers
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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 ...
Ramin's user avatar
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