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4
votes
3answers
403 views

Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares? Be it 1,2,3,...n or k,k+1,k+2,...n.
3
votes
0answers
128 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 ...
3
votes
4answers
388 views

solutions to special diophantine equations [closed]

Let $0\le x,y,z,u,v,w\le n$ be integer numbers obeying \begin{align*} x^2+y^2+z^2=&u^2+v^2+w^2\\ x+y+v=&u+w+z\\ x\neq& w \end{align*} (Please note that the second equality is ...
9
votes
1answer
167 views

Schoenberg's Rational Polygon Problem

"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by ...
23
votes
4answers
4k views

Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
4
votes
1answer
311 views

A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$

Is the following conjecture true? Conjecture. If $r > s \ge 1$ are relatively prime integers such that \begin{equation} (r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1} \end{equation} then $r-s = 1$ ...
0
votes
3answers
176 views

Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?

In this MSE question/thread, I have been discussing the equation $$ (x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$} $$ where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to ...
11
votes
4answers
2k views

hard diophantine equation: $x^3 + y^5 = z^7$

Does the equation $x^3+y^5=z^7$ have a solution $(x,y,z)$ with $x,y,z$ positive integers and $(x,y)=1$? In his book H. Cohen (Number theory,2007) said "[...] seems presently out of reach". I couldn't ...
3
votes
0answers
120 views

polynomials in many variables and Hasse principle

I was wondering whether there exists any result of the form "if $f \in \mathbb{Z}[x_1, ..., x_k]$ is a polynomial (not form! I don't require homogeneity) of total degree $n$, with $k \geq \delta ...
11
votes
2answers
561 views

Failing of heuristics from circle method

The heuristic from circle method for integral points on diagonal cubic surfaces $x^3+y^3+z^3=a$ ($a$ is a cubic-free integer) seems to fit well with numerical computations by ANDREAS-STEPHAN ELSENHANS ...
1
vote
1answer
959 views

The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
12
votes
3answers
1k views

Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$

Is the following conjecture correct? Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < ...
0
votes
0answers
75 views

Superelliptic Curves [duplicate]

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...
7
votes
1answer
257 views

The Diophantine equation $x^p - 4y^p = z^2$

If $p \geq 5$ is a prime, are there any integers $x, y, z > p$ such that $(x, y) = 1$ and $$x^{p} - 4y^{p} = z^{2}$$
1
vote
0answers
292 views

When is a cubic polynomial a cube? [closed]

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
0
votes
1answer
543 views

Is surface $x^2+z^2=2\cdot y^2$ something of a Möbius strip?

This question is naive. My association with Möbius strip comes from not being able to smoothly extract positive solutions of the diophantine equation $$x^2+z^2=2\cdot y^2$$ I got a parametrization ...
7
votes
3answers
803 views

How many integer points does my favorite ellipse go through?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ x^2+y^2 - ...
-4
votes
2answers
188 views

If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$? [closed]

If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$? I think this is true, how to prove this?
15
votes
0answers
427 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
3answers
956 views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
2
votes
1answer
242 views

Integer points on $y^2=x^2-x^3+x^4$

Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than $x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, ...
15
votes
0answers
252 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 ...
3
votes
1answer
449 views

Some types of diophantine equations and their decidability

The MDRP theorem – which answers Hilbert's tenth problem in the negative – says: There is no algorithm for determining whether an arbitrary diophantine equation has a solution. In ...
4
votes
1answer
462 views

Hyperrectangles with integer diagonals

What is the largest value of $n$ for which there exists $n$ (not necessarily distinct) complete squares of natural numbers such that the sum of every subset of it is also a complete square? ( For ...
5
votes
4answers
441 views

seeking an integer parameterization for A^2+B^2=C^2+D^2+1

I'm looking for a complete [integer] parameterization of all integer solutions to the Diophantine equation $A^2+B^2=C^2+D^2+1$, analogous to the classical parameterization of the Pythagorean ...
1
vote
0answers
126 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
1answer
143 views

Link between integral points on varieties and solutions to Diophantine equations

Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X_1,\dots,X_n)$ a polynomial in $k[X_1,\dots,X_n]$. I am looking for notes, books or surveys detailing ...
12
votes
1answer
548 views

Can we extend the proof of Catalan's conjecture?

What is it, in Mihailescu's proof of Catalan conjecture, that uses explicitly the fact that there is a 1 on the right hand side of $x^p - y^q = 1$? In other words, why can't we extend his argument to ...
2
votes
2answers
178 views

Enumerating solutions to an underdetermined non-homogenous linear system of Diophantine equations

I have a large, under-determined system (60 equations and 116 unknowns) of linear Diophantine equations. I am aware of the algorithms typically used to solve these systems, which is not my question. ...
2
votes
0answers
124 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
1answer
396 views

Diophantine equations with infinitely many large solutions

Let $F(x,y)$ be a squarefree binary form with integer coefficients, possibly reducible, $\deg(F) \ge 3$. I am interested in ways of getting infinitely many integer solutions $(x,y,m), m \ne 0$ to ...
1
vote
0answers
40 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 ...
5
votes
1answer
264 views

$xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3$ in nonvanishing integers

From research completely unrelated to Number Theory I stumbled onto the following equation: $$ xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3 $$ for $x, y, z$ integers, $x,y,z \neq 0$. Are ...
3
votes
2answers
441 views

A remark of Mordell alluding to a local/global principle for cubic Diophantine equations

In Mordell Diophantine Equations he says: In recent years it has been shown that there seems to be a close connection between the number of solutions of f(x,y) = 0 (mod $p^r$) and the existence of ...
9
votes
3answers
2k views

Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$

I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are \begin{equation} (r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, ...
1
vote
0answers
53 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). ...
9
votes
3answers
6k views

Status of Beal, Granville, Tijdeman-Zagier Conjecture [closed]

The Beal, Granville, Tijdeman-Zagier Conjecture, i.e. If $A^x+B^y=C^z$ , where $A, B, C, x, y,z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ must have a ...
1
vote
2answers
250 views

Computational complexity of solution of Pell equation and more

What is computational complexity for computing integral solution of Pell equation .It seems to be in P ,and could any one give an algorithm and reference for proof of it's complexity? And more,could ...
3
votes
0answers
138 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 ...
0
votes
0answers
96 views

Are there generating functions of rational or integral solutions of Diophantine equation that

As we know,there are generating functions for c.e.languages which are some retricted rational or algebraic or transcendental functions dependent on the class of languages like regular ...
0
votes
0answers
78 views

Using the circle method to prove that there are no solutions to diophantine equaltions

Would it be possible to use the circle method to prove that there are no solutions to certain diophantine equations. For example, could one use the circle method to prove the fact that there are no ...
2
votes
0answers
175 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
1answer
184 views

Non-coprime solutions to x^n+y^n = z^2

Let $n$ be an odd prime. I know that the equation $x^n+y^n = z^2$ has no non-zero coprime solution in integers whenever $n \geq 5$, and that there are infinitely many solutions as soon as one drops ...
11
votes
3answers
297 views

(Non-)Existence of curves of low degree on affine and projective varieties

I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...
2
votes
3answers
224 views

Specific Diophantine Equation Appearing in Faa Di Bruno Formula

In a Faa Di Bruno Formula there is an equation: $m_1$+2*$m_2$+3*$m_3$+...n*$m_n$=n Is there any general solution for this equation. For example for $m_1$+$m_2$+$m_3$+...+$m_n$=n, there is a ...
1
vote
1answer
217 views

On $x^3-y^2=1728 \text{ unit}$ in number fields

Consider solution of $$x^3-y^2=1728 \text{ unit} \qquad (1)$$ in a number field. This is related to the discriminant of elliptic curve in terms of $c_4,c_6$. Via elliptic curves it might have ...
2
votes
0answers
231 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 ...
5
votes
4answers
2k views

General integer solution for $x^2+y^2-z^2=\pm 1$

How to find general solution (in terms of parameters) for diophantine equations $x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$? It's easy to find such solutions for $x^2+y^2-z^2=0$ or $x^2+y^2-z^2-w^2=0$ or ...
0
votes
1answer
330 views

Erdős-Straus with 4 terms

The Erdős-Straus conjecture states that any fraction of the form $\frac{4}{n}$ can be decomposed as an Egyptian fraction with just 3 terms. In related research, I've recently come across conditions on ...
10
votes
7answers
3k views

Is there an algorithm to solve quadratic Diophantine equations?

I was asked two questions related to Diophantine equations. Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all ...