The tag has no wiki summary.

learn more… | top users | synonyms

14
votes
0answers
381 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 ...
10
votes
0answers
780 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 ...
9
votes
0answers
582 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 ...
9
votes
0answers
304 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 ...
7
votes
0answers
215 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
0answers
426 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
0answers
152 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
0answers
194 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
0answers
216 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
0answers
266 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
0answers
89 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 ...
3
votes
0answers
99 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 ...
3
votes
0answers
189 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
0answers
118 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
0answers
245 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 ...
3
votes
0answers
215 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
0answers
86 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
0answers
312 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
0answers
343 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
0answers
107 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
0answers
90 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
0answers
68 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 ...
2
votes
0answers
218 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
0answers
492 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 ...
2
votes
0answers
206 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
0answers
191 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
0answers
231 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 ...
2
votes
0answers
485 views

Does the following Diophantine equation have nontrivial rational solutions?

Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all? If I am ...
1
vote
0answers
144 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 ...
1
vote
0answers
190 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
0answers
128 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 ...
1
vote
0answers
363 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
0answers
160 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
0answers
289 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. ...
1
vote
0answers
164 views

Checking local solubility of varieties at “bad” primes

Let $X$ be smooth variety defined over $\mathbb{Q}$. If we want to check that $X$ is locally soluble at a prime $p$, then it suffices to find a non-singular $\mathbb{F}_p$ point, which can be lifted ...
0
votes
0answers
100 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 ...
0
votes
0answers
129 views

On unique solutions to linear diophantine equations

Let $\sum_{i=1}^k a_ix_i = N$ be the equation with $a_i \in [2^t,2^{t+1}]$ being distinct primes. If we seek unique solutions $x_i\in R_i = (0,a_i)\cap \Bbb Z$, then in general it is not possible. ...
0
votes
0answers
504 views

Diophantine: a^n + b^n + c^n = d^n and a^n + b^n = c^n + d^n

Let us consider the equation $a^n+b^n=c^n$ for positive integers $a,b,c$ and $n\ge 2$. The $n=2$ case has a well-known and beautiful parametrization known as Pythagorean triples. Fermat's Last Theorem ...
0
votes
0answers
320 views

Diophantine equation solutions

I am not able to make headway on solving the diophantine equation $x^m - y^n = 6.$ Are there any solutions to this? What about $x^m - y^n = 14,$ and $= 30$ (both $m$ and $n$ are at least $2$).
0
votes
0answers
90 views

how do you bound exponent of x^2+1=y^p

for p a prime exponent using linear forms in logs? So far I have (x-i)(x+i)=y^p which are coprime and hence x+i=(a+ib)^p , now how do I get a linear form in logs so that I can find an upper bound on ...
0
votes
0answers
544 views

algorithm for solving systems of linear Diophantine inequalities

So, I posted on stack overflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem ...
0
votes
0answers
262 views

Exponential Diophantine $\prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$ ,$e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$

Does the exponential diophantine equation $$ \prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$$ with $x_i, y_i,z_i$ coprime and $e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$ have ...
0
votes
0answers
269 views

When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?

It is easy to show that the following problems are equivalent. a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square? and b. When is $X^2-PY^2=k$ ...