# Tagged Questions

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**18**

votes

**0**answers

322 views

### Are there any integers which can't be written as a sum of two fourth powers minus a cube?

To be precise, I am asking:
Does there exist an integer $k$ such that there do not exist (possibly negative) integers $x,y,z$ satisfying $x^4+y^4=z^3+k$?
Heuristically the answer must be yes, in ...

**15**

votes

**0**answers

457 views

### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with ...

**14**

votes

**0**answers

430 views

### Torsion points of abelian varieties in the perfect closure of a function field

The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...

**11**

votes

**0**answers

427 views

### Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...

**10**

votes

**0**answers

912 views

### Consecutive averages of sequence (or difference quotients of partial sums) always square

I proposed the following problem for the December 2013 USA IMO TST earlier this month:
Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s ...

**10**

votes

**0**answers

328 views

### Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?

Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for ...

**10**

votes

**0**answers

864 views

### Effective proofs of Siegel's theorem using arithmetic geometry

This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is ...

**7**

votes

**0**answers

195 views

### Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being
$x^2=4+8y^2+13z^2$.
The ideal answer would be a way to parametrize all the integer ...

**7**

votes

**0**answers

232 views

### When adding a constant makes a multivariate polynomial reducible?

Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees?
...

**7**

votes

**0**answers

477 views

### Diophantine $x^p+y^q=(x+y)^r$

Is the equation:
$$x^p+y^q=(x+y)^r$$
in integers $x,y,z,p,q,r$ with $p \geq 2,q \geq 2, r \geq 2$ complete solved?
For $(p,q,r)=(n,n,n+1)$ a parametrization is $t=1-s$ and $ ...

**5**

votes

**0**answers

197 views

### When does the Lloyd polynomial have only integral roots?

For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by
$$
L_t(n,x):=\sum_{j=0}^t(-1)^j\binom{x-1}{j}\binom{n-x}{t-j}(q-1)^{t-j}.
$$
A ...

**4**

votes

**0**answers

173 views

### Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes
as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either
returns true or false, we say that $\mathcal{A}$ works for ...

**4**

votes

**0**answers

106 views

### Integers in a given box that can be represented by a polynomial

Suppose that $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ is a polynomial of degree $d$, and examine the quantity
$$\displaystyle N(F;X, B) = \# \{(x_1, \cdots, x_n) \in \mathbb{Z}^n | -X ...

**4**

votes

**0**answers

105 views

### Curves on hypersurfaces generated by diagonal sums

This is related to an earlier question of mine ((Non-)Existence of curves of low degree on affine and projective varieties). It seems that the question is too difficult for specific surfaces, although ...

**4**

votes

**0**answers

232 views

### Counting Special Rational Points on Cubic Surfaces

A paper of Heath-Brown gives an heuristic argument for the density of rational points on two cubic surfaces: $x^3+y^3+z^3=kw^3,k=2,3$, say, the number of rational points of height less than $N$ on ...

**4**

votes

**0**answers

303 views

### $a^5+b^5=c^5+d^5$ and polynomial identities

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from genus 0 curve ...

**4**

votes

**0**answers

238 views

### Expressions of $tanh$ type whose continued fractions have two shifts per period

This is a follow-up of another thread about quasi periodic continued fractions, a.k.a. Hurwitz fractions, with some linear shifts. I seem to have found the pattern of a subclass of them, as given ...

**4**

votes

**0**answers

291 views

### A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...

**3**

votes

**0**answers

72 views

### Low height integer points on a rank variety

Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition
$$
\mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1.
...

**3**

votes

**0**answers

241 views

### On 7th and 8th powers for $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$

The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for either $k=5$ or $6$ is quite well explored, and it has long been known that it has an infinite number of primitive ...

**3**

votes

**0**answers

147 views

### A question on a paper by Ribet

I'm reading the article On the equation $a^p + 2^\alpha b^p + c^p = 0$ by Ribet (http://math.berkeley.edu/~ribet/Articles/acta.pdf), but I'm having trouble understanding his proof of Theorem 3. For ...

**3**

votes

**0**answers

286 views

### Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions ...

**3**

votes

**0**answers

218 views

### Ternary form related to identity for abc conjecture

Consider the identity:
$$ \begin{aligned}
f_1 &= 4 (4 x + z) \cdot z^{3} \\
f &= x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4} + 4 x^{3} z + 12 x^{2} y z + 12 x y^{2} z + 4 y^{3} z + ...

**3**

votes

**0**answers

130 views

### Effective Lang-Weil bounds for del Pezzo surfaces

Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$.
By the Lang-Weil bounds
$ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ ...

**3**

votes

**0**answers

264 views

### quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ...

**3**

votes

**0**answers

91 views

### Weak classes of diophantine functions

From a well-known work(s) by Putnam, Davis, Robinson and Matiyasevich, we know that every partially recursive function is diophantine.
Now it seems a natural question to ask: can we say something ...

**3**

votes

**0**answers

854 views

### 0,1 solution to system of linear integer equations.

I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-maxtrix and $m$-vector of nonnegative intgers (respectivelly).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...

**3**

votes

**0**answers

328 views

### The surface $ x^2 y^2 + 1 = (x^2 + y^2) z^2 $

Hi, I'm trying to find all rational points on the surface of the title, in connection with the Euler Brick (AKA Rational Box) problem.
This surface is equivalent to $ x^2 z^2 - 1 = (x^2 - z^2) y^2 $, ...

**3**

votes

**0**answers

366 views

### Asymptotics related to the Erdos--Moser diophantine equation

I share the authorship of this question with Pieter Moree.
In our recent joint work with Y. Gallot (arXiv:0907.1356 [math.NT]) we attack
the Erdős--Moser diophantine equation
$$
...

**2**

votes

**0**answers

162 views

### Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$
I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that
...

**2**

votes

**0**answers

148 views

### Diophantine equations and the numbers $4,7,8$

Consider the diophantine equation
$$
x^n+y^n+z^n=k\cdot xyz,
$$
where $n,x,y,z$ are positive integers. Now consider $k\in\left\{4,7,8\right\}$. It is well-known or easily provable that for $n=1$ and ...

**2**

votes

**0**answers

199 views

### Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...

**2**

votes

**0**answers

136 views

### n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...

**2**

votes

**0**answers

129 views

### What is the general feeling for Hilbert's 10th problem for Q?

We know that Hilbert's 10th problem for $\mathbb{Z}$ is undecidable. I was wondering whether there is a strong opinion in the mathematical community on the decidability of Hilbert's 10th for ...

**2**

votes

**0**answers

107 views

### Congruences of binary forms

Suppose $f(x,y) \in \mathbb{Z}[x,y]$ is a binary form (that is, homogeneous polynomial in two variables). Further suppose that $f$ is irreducible over $\mathbb{Z}$ and has no fixed prime divisor. Let ...

**2**

votes

**0**answers

237 views

### Algorithm for solutions to quadratic forms over number fields

Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)?
I am especially interested in the quaternary case. There exist some ...

**2**

votes

**0**answers

138 views

### Reference for original paper (but translated to English) of Matiyasevich's proof of Fibonacci relation being Diophantine?

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems that is accessible to ...

**2**

votes

**0**answers

215 views

### Hurwitz integers and $F_4$

The Hurwitz integers are
$$
\mathcal H=
\{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.
$$
I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...

**2**

votes

**0**answers

195 views

### is exponential diophantine over Qp

Thanks to Matiyasevic, we all know that exponential is diophantine over the integers. Also, thanks to transcendental number theory, we know that exponential is not diophantine over the rationals. So ...

**2**

votes

**0**answers

274 views

### Prove a parametrization function is surjective

As a starting note, I would like to say that I haven't (yet) taken courses in Set Theory, so some higher-level notation may be lost on me (and I may not write everything conventionally), but I'll do ...

**1**

vote

**0**answers

450 views

### Is $ a^2b^2-ab-ap$ a perfect square for suitable $a,b\in \mathbb{Z}^+$?

Consider the expression
$$
a^2b^2-ab-ap\qquad (a,b\in \mathbb{Z}^+),
$$
where $p\equiv1\pmod{4}$.
Question. For every prime $p\equiv1\pmod{4}$ do there exist $a,b\in\mathbb{Z}^+$ such that ...

**1**

vote

**0**answers

136 views

### Equation in the Gaussian Integers

Let $a,b \in \mathbb{N}$. Is there a possibility to characterize the solutions of $a N(\alpha) - b N(\beta)=1$ where $\alpha,\beta \in \mathbb{Z}[i]$? In particular I am interested in the case $a=1$ ...

**1**

vote

**0**answers

43 views

### Question about link between non-terminals of grammars and variables of Diophantine equations

If we change the right arrow in the rewriting rules of grammar into equators , changes all terminals into x and keep the non-terminals unchanged,we get system of equations.In some cases,those ...

**1**

vote

**0**answers

77 views

### Cassels-Birch-Davenport theorem for multiple quadratic forms of certain type

A classical theorem of Cassels states that if a homogenous quadratic form $Q$ has an integer zero, then there is a zero of small height (bounded solely by the coefficients and number of variables). ...

**1**

vote

**0**answers

207 views

### Integral points on affine rational curves over $\mathbb{Q}$

Given a rational curve $C:(f_1(t),f_2(t))$, where $f_i(t),i=1,2$ are rational functions with rational coefficients.
Question: Is there any criterion(proved or conjectural) for the existence of ...

**1**

vote

**0**answers

106 views

### Rational solutions of equations of the form $y^2 x = f(x)$

Let $k$ be any number field, and suppose we want to study the $k$-rational points on
$$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a ...

**1**

vote

**0**answers

84 views

### Is there any track for proving $D=NP$, besides showing that $D$ has polynomial-bounded universal quantifiers?

Background
By the MRDP theorem, every for every recursively enumerable set $S$, there exists a Diophantine polynomial $p$ such that
$$x \in S \iff \exists y_1, \dots, y_n \in \mathbb{N} \text{ such ...

**1**

vote

**0**answers

395 views

### Diophantine equation over Z[i]

I'm trying to generate the set of solutions of a specific diophantine equation over Z[i].
The equation is the following:
$$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$
with $ z_1$ (resp $z_2$) such that ...

**1**

vote

**0**answers

172 views

### Efficient counting of Egyptian fractions with bounded denominators

I was amazed to discover that sequence http://oeis.org/A020473 in the OEIS has almost four hundred terms computed.
I wonder how one can get that far? E.g., how one can compute A020473(100)?
P.S. ...

**1**

vote

**0**answers

307 views

### Quartic case of a theorem of Bombieri and Pila

I am interested in the ternary case of a theorem of Bombieri and Pila, in E. Bombieri and J. Pila, "The number of integral points on arcs and ovals", Duke Mathematical Journal., 59 (1989), 337-357. ...