3
votes
0answers
126 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
387 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 ...
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 ...
4
votes
1answer
310 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 $r-s = 1$ ...
11
votes
2answers
560 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 ...
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 < ...
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 = ...
-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 ...
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: $x=3,y=15$, $x=5,y=55$, ...
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 ...
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 ...
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 ...
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
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 ...
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 ...
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
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). ...
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 ...
1
vote
1answer
216 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 ...
21
votes
3answers
665 views

Consecutive square values of cubic polynomials

Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square? It is known that the ...
9
votes
1answer
442 views

Integer Solutions of $x+y^n = y + x^m$ for $n < m$

I found 8 of them and believe there is no more: $$2+3^2=3+2^3$$ $$2+6^2=6+2^5$$ $$6+15^2=15+6^3$$ $$3+16^2=16+3^5$$ $$3+13^3=13+3^7$$ $$2+91^2=91+2^{13}$$ $$5+280^2=280+5^7$$ $$30+4930^2=4930+30^5$$ ...
3
votes
0answers
97 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 ...
6
votes
4answers
290 views

Number of solutions of linear homogenous Diophantine equation inside a box

Let $a_1, ..., a_d$ be positive reals and consider the linear Diophantine equation $$ \sum_i a_in_i = 0. $$ I am interested in estimating the number of integer solutions of this equation inside a ...
5
votes
1answer
406 views

Is the following consequence of the Lang conjecture known?

This came up in a discussion with a colleague of mine, who studies PDEs. He was asking for a function $f \colon \mathbb{N} \rightarrow \mathbb{N}$ such that, for all but finitely many $n$, the ...
3
votes
0answers
100 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 ...
5
votes
1answer
286 views

Subsets of all Diophantine's sets

I have asked this question on math.stackexchange already: http://math.stackexchange.com/questions/627461/subsets-of-all-diophantines-sets Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable ...
2
votes
0answers
101 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 ...
3
votes
1answer
186 views

Special Case of famous Equation

I'm interested in the following diophantine eqaution: $(5^n-1)/4=y^2$. It turns out that this is a special case of the Nagell-Ljunggren equation, where $x=5$ and $q=2$ It has been shown that for ...
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 ...
0
votes
1answer
157 views

Reference request: on sums of the form $ax^m + by^n = h$

I know that equations of the form $$\displaystyle ax^d + by^d = h$$ with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
10
votes
0answers
785 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 ...
6
votes
3answers
425 views

On the equation $a^n + b^n = c^2$

I am interested in the possible natural solutions of the equation $a^n + b^n = c^2$ where $n \geq 4$ is fixed. I am not sure if it is well-known or not, so any suggestion would be helpful.
10
votes
1answer
387 views

Can the sum of two non-zero coprime fifth powers be powerful?

I am wondering if the sum of two non-zero coprime fifth powers can be powerful. There are no small solutions. Q1 Can the sum of two non-zero coprime fifth powers be powerful? Got a partial ...
6
votes
2answers
433 views

The Theory of Transfinite Diophantine Equations [closed]

The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
13
votes
1answer
453 views

State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations $$a^5 + b^5 = c^5 + d^5$$ and $$a^6 + b^6 = c^6 + d^6$$ have no known nontrivial solutions, but $$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$ and ...
2
votes
1answer
138 views

Integral values of rational map

This question is related to this post on Math.MO. A theorem of B.Segre tells us that if there is one rational point on a non-singular cubic surface $X$ over $\mathbb{Q}$, then the surface is ...
-4
votes
1answer
141 views

$p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
4
votes
0answers
213 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 ...
2
votes
1answer
133 views

Paired Quadratic diophantine equations

For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$ If so, how to ...
3
votes
0answers
202 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 + ...
4
votes
2answers
328 views

On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$

Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the ...
7
votes
2answers
493 views

The equation $x^m-1=y^n+y^{n-1}+…+1$ in prime powers $x,y$

Does the equation $x^m-1=y^n+y^{n-1}+...+1$ have only finitely many solutions $(x,y,m,n)$ where $x,y$ are prime powers with $y>2$ and $m,n$ are integers with $m,n>1$? This question arose in the ...
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 - ...
11
votes
2answers
674 views

On Generalizations of Fermat's Conjecture

We know the following facts: (1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$. (2) For all $3\leq n$ the equation ...
3
votes
1answer
341 views

Proving conditions on $(r+s)^2 \mid (4r^4+1)$, related to Pell oblongs

While working on another problem (Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$), I came across a question which seems to be of [semi-] independent interest. Conjecture. If ...