Questions tagged [diophantine-equations]

Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...

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The (last step of the) proof that the set of badly approximable matrices has measure zero

An $m \times n$ matrix $A$ is called badly approximable if there exists $c > 0$ such that for all integer vectors $p \in \mathbb Z^m$ and $q \in \mathbb Z^n-\{0\}$ we have $$ \|A q + p \| \ge c \| ...
No One's user avatar
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Integers representable as binary quadratic forms

It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...
Bogdan Grechuk's user avatar
8 votes
2 answers
686 views

Solving functional equation $f(xy)=f(x+y)$ and Diophantine equations

It is well known that the only solution is $f$ a constant function. However, by putting some restrictions on the functional equation, we might get other solutions, with potential implications to ...
Vincent Granville's user avatar
1 vote
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4-distance problem and elliptic curves

The 4-distance problem is an open question(as far as I know it is still open) that asks if there exists a point P on the Euclidean plane such that its distances to all four points of a unit square are ...
Yuan Yang's user avatar
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Can $12n+5$ be written as $2x^2+5y^2+9z^2+xyz$ with $x,y,z$ nonnegative integers?

Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, for any $n\in\mathbb N$ we can write $4n+1$ as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Motivated by this, here ...
Zhi-Wei Sun's user avatar
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Can $y^2-4$ be a divisor of $x^3-x^2-2 x+1$?

Do there exist integers $x$ and $y$ such that $\frac{x^3-x^2-2 x+1}{y^2-4}$ is an integer? In other words, can any integer representable as $x^3-x^2-2 x+1$ have any divisor representable as $y^2-4$? ...
Bogdan Grechuk's user avatar
1 vote
1 answer
165 views

Fundamental solutions to linear Diophantine equations and their existence and computation

$T>0$ is a parameter. Consider the linear Diophantine equation $ax+by=c$ where $a,b$ are coprime. Suppose $a,b$ are of magnitude $T^{1+\epsilon}$ and $c$ is of magnitude $T^2$. For how many such ...
Turbo's user avatar
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Why is Hilbert’s 11th problem still partially resolved?

Hilbert’s 11th problem which demands that we ‘classify quadratic forms over algebraic number fields’ has been of interest to me and I would like to know what makes it partially resolved currently. Or ...
A. J.'s user avatar
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Can each natural number be represented by $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?

It is well known that each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $2w^2+x^2+y^2+z^2$ with $w,x,y,z\in\mathbb N$. Furthermore, $$\{2w^2+x^2+y^2:\ w,x,y\in\mathbb N\}=\mathbb N\setminus\{4^k(...
Zhi-Wei Sun's user avatar
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2 votes
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What are the integer solutions to $y^3=2x^3+x+1$?

The question is in the title. Short motivation. Consider Diophantine equations in $2$ variables. Quadratic ones are easy, and can be solved, for example, here https://www.alpertron.com.ar/QUAD.HTM. ...
Bogdan Grechuk's user avatar
2 votes
0 answers
232 views

Solving $x^k+y^k+z^k=w^k$ non-trivially in strictly positive integers

Consider the equation $x^k+y^k+z^k=w^k$ in $x$, $y$, $z$ and $w$ with $k\in\mathbb{N}_{\geq2}$. If we look for solutions that are strictly positive and non-trivial i.e. $x\neq-y$, $x\neq w$ etc... ...
Ivan Meir's user avatar
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Reference request: "A result of Siegel" related to Ramanujan-Nagell type equations

Wikipedia refers to the Diophantine equation $ x^2 + D = AB^n $ as an "equation of Ramanujan–Nagell type". It also says that "A result of Siegel implies that the number of solutions in ...
Eric Nathan Stucky's user avatar
1 vote
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Jones–Sato–Wada–Wiens diophantine equation [closed]

I came across this from the 1993 book Matiyasevic - Hilbert's 10th problem. Typeset from another question: \begin{align} P(a,b,\dotsc,z)=(k+2)\Bigl(1&-(wz+h+j-q)^2\\ &-\left[(gk+2g+k+1)...
abrahimladha's user avatar
1 vote
1 answer
177 views

On integral points of $f(x,y)=z g(x,y)$

Let $f(x,y),g(x,y)$ be polynomials with integer coefficients. Consider the surface $$ f(x,y)=z g(x,y) \qquad (1)$$ (1) has parametrization over the rationals given by $z=\frac{f(x,y)}{g(x,y)}$. Q1 ...
joro's user avatar
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10 votes
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262 views

$y^3=x^4+x+1$, and rational points on rank 2 Picard curves

What are (a) integer, (b) rational solutions to the equation $$ y^3 = x^4 + x + 1 ? $$ There are obvious solutions $(x,y)=(-1,1)$ and $(0,1)$, are they the only ones? Context: There are a lot of ...
Bogdan Grechuk's user avatar
-4 votes
1 answer
196 views

Does the equation $x^k+y^k-z^k-w^k=3\ (k>3)$ have a solution over $\mathbb N$?

Clearly, $$3=0^2+2^2-1^2-0^2\ \ \mbox{and}\ \ 3=4^3 +4^3-5^3-0^3.$$ Question. Let $k>3$ be an integer. Does the equation $$ x^k+y^k-z^k-w^k=3\quad \ (x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})\tag{1}$$ ...
Zhi-Wei Sun's user avatar
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Can the equation $n=x^6-y^6+z^3-w^3$ with $x,y,z,w\in\mathbb Q_{\ge0}$ be solved via an identity?

Let $\mathbb Q_{\ge0}$ denote the set of all nonnegative rational numbers. In 1923 Richmond proved that each $r\in\mathbb Q_{\ge0}$ can be written as $x^3+y^3+z^3$ with $x,y,z\in\mathbb Q_{\ge0}$. In ...
Zhi-Wei Sun's user avatar
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14 votes
1 answer
495 views

Does $A-A=\mathbb Q$ hold for $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$?

Let $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$. Then $$A-A:=\{a-b:\ a,b\in A\}=\{u^4+v^4-x^4-y^4:\ u,v,x,y\in\mathbb Q\}.$$ Motivated by Question 415482, here I ask the following question. Question. Is it true ...
Zhi-Wei Sun's user avatar
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3 votes
1 answer
281 views

Waring's problem over $\mathbb Q_{\ge0}$

Let $k$ be a positive integer. Note that $a/b=ab^{k-1}/b^k$ for any integers $a$ and $b>0$. If every $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x_1^k+\cdots+x_{s}^k$ with $x_1,\ldots,x_s\...
Zhi-Wei Sun's user avatar
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3 votes
0 answers
385 views

Simple Diophantine equation

Are there any solutions in positive integers of $x^3 + 1 = (x - k) y^3$? The closest I can get is $19^3 + 1 = 20 \times 7^3$, but $20\gt 19$ so it just misses! For the related $x^3 - 1 = (x - k) y^3$,...
Joe Shipman's user avatar
3 votes
1 answer
557 views

Number of points of a quadric hypersurface over a finite field

Let $k = \mathbb{F}_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n_k$ a quadric hypersurface defined over $k$. By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is ...
Puzzled's user avatar
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11 votes
0 answers
362 views

Is it true that $\{x^4+y^3+z^2:\ x,y,z\in\mathbb Q_{\ge0}\}=\mathbb Q_{\ge0}$?

Let $\mathbb Q_{\ge0}$ be the set of all nonnegative rational numbers. I have the following conjecture based on my computation. 4-3-2 Conjecture. Each $r\in\mathbb Q_{\ge0}$ can be written as $x^4+y^3+...
Zhi-Wei Sun's user avatar
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1 vote
0 answers
485 views

How to describe all integer solutions to $x^2+y^2=z^3+1$?

The question is to find all integer solutions to the equation $$ x^2+y^2=z^3+1. $$ This equation obviously has infinitely many integer solutions (take, for example, $(x,y,z)=(1,u^3,u^2)$ for any ...
Bogdan Grechuk's user avatar
10 votes
3 answers
736 views

Solve in integers: $y(x^2+1)=z^2+1$

Find all integer solutions to the equation $$ y(x^2+1)=z^2+1. $$ There is, for example, an infinite family of solutions $x=u$, $y=(uv\pm1)^2+v^2$, $z=(u^2+1)v \pm u$, $u,v \in {\mathbb Z}$, but there ...
Bogdan Grechuk's user avatar
5 votes
0 answers
282 views

On $w^4+x^4+y^2+z^2$ over a number field

In 1921 Siegel confirmed a conjecture of Hilbert by proving that for any number field $K$ each element of $$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
Zhi-Wei Sun's user avatar
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2 votes
1 answer
269 views

Diophantine equations and ergodic theorems

In the paper by Akos Magyar, Diophantine Equations and Ergodic Theorems, one states in page 923 the following theorem: Theorem 1: Let $Q(m)$ be a nondegenerate polynomial and $\Lambda$ is ...
Elmustapha NADIR's user avatar
16 votes
2 answers
820 views

Are there infinitely many positive integer solutions to $(3+3k+l)^2=m\,(k\,l-k^3-1)$?

I usually work in the field of differential geometry, but I have encountered the following problem in my research: Are there infinitely many positive integers $k,l,m\in\mathbb N^{>0}$ such that $$(...
Sebastian's user avatar
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3 votes
1 answer
832 views

Find all integer solutions to the following easy-looking Diophantine equations

In general, it is not clear What does it mean to solve an equation? in integers. In this question, let us assume that an equation $$ P(x_1,\dots,x_n)=0 $$ is solved if we have proved that its integer ...
Bogdan Grechuk's user avatar
0 votes
1 answer
118 views

Special type of normal form of matrix in principal ideal domain

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric. Can one always ...
en kuo's user avatar
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4 votes
1 answer
521 views

About a result in Martin Davis' 1973 article "Hilbert's Tenth Problem is Unsolvable"

In Martin Davis, Hilbert's Tenth Problem is Unsolvable, The American Mathematical Monthly, Vol. 80, No. 3 (Mar., 1973), pp. 233-269 (link), the author prove the following result: Theorem 3.1: For ...
Safwane's user avatar
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Prove there are infinitely many squares which are the sum of two tetrahedral numbers [closed]

Let $T_n = \frac{1}{6}n(n+1)(n+2)$ denote the $n$th Tetrahedral number. The first several solutions to squares as sums of two Tetrahedral numbers are {T_n,T_m,a^2} 1 5 6\ 1 8 11\ 1 22 45\ 1 24 51\ 1 ...
Benjamin L. Warren's user avatar
5 votes
6 answers
3k views

How many cubes are the sum of three positive cubes?

Are there infinitely many integer positive cubes $x^3 = a^3 + b^3 + c^3$ that are equal to the sum of three integer positive cubes? If not, how many of them are there?
user473935's user avatar
3 votes
3 answers
531 views

Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$

If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the equation $$ yz=...
Bogdan Grechuk's user avatar
1 vote
1 answer
139 views

On parametrization of a type of unimodular $2\times2$ integral matrix

A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}\in\mathbb Z^{2\times 2}$ is unimodular if $$|wz-xy|=1$$ holds. Is there a parametrization of such matrices with $|w||z|-xy=1$ $$w,z<0<\max(...
Turbo's user avatar
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23 votes
1 answer
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On the smallest open Diophantine equations: beyond Hilbert's 10 problem

In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all ...
Bogdan Grechuk's user avatar
4 votes
0 answers
299 views

What does it mean to solve an equation?

Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation $$ P(x_1,\dots,x_n) = 0 $$ where $P$ is a polynomial with integer coefficients. Do we have a ...
Bogdan Grechuk's user avatar
16 votes
2 answers
1k views

Representing $x^3-2$ as a sum of two squares

Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers. Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
Bogdan Grechuk's user avatar
7 votes
0 answers
230 views

Magic hourglass of squares hyperelliptic equation

I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so: $a^2$ $b^2$ $c^2$ $ $ $ $ $ $ $ $ $ $ $d^2$ $e^2$ $f^2$ $g^2$ ...
Thomas's user avatar
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The strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation

Let us consider the strong twin conjecture: For all positive integer $n$ there exist a prime $p$ such that $$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime Since the inequalities and the ...
Safwane's user avatar
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7 votes
1 answer
419 views

$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves

Do there exists rational numbers $x$ and $y$ such that $$ y^3 = x^4 + x + 2 ? $$ Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
Bogdan Grechuk's user avatar
0 votes
0 answers
187 views

conjecture about quadratic equation

All the variables I deal with are integers. Given $q \geq 2, |k| \geq 2$, impose a constraints that we have an integer $p$ such that $q^2=1+pk=1\pmod k.$ For example, $3^2=1+1\cdot8, 5^2=1+3\cdot8.$ ...
En-Jui Kuo's user avatar
2 votes
0 answers
243 views

Integer points on genus 1 curves using CAS

How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.? As a specific example, do ...
Bogdan Grechuk's user avatar
1 vote
0 answers
74 views

Conjectures about the automorphism group of integer lattice by enlarging the matrix

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Notation: $\GL(n, \mathbb{Z})$ to be the set of $n \times n$ invertible matrix, and ...
en kuo's user avatar
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5 votes
0 answers
324 views

Does the equation $a^b+b^c+c^a=d^e$ have solutions in $\mathbb {N}$

Here $a,b,c,d,e$ are distinct and all greater than $1$. This question was formerly posted on Math.Stackexchange, precisely here, but seems to be more general than some other tough number theory ...
Derek Luna's user avatar
1 vote
0 answers
89 views

Diophantine equation about the automorphism group of lattice by constraints

Fixed $\sigma_x=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right)$ and $K=\left( \begin{array}{ccc} 3 & 32 & -64 \\ 1 & 32 & -32 \\ -2 & -32 & 64 \\ \...
En-Jui Kuo's user avatar
8 votes
0 answers
232 views

Hilbert 10th problem for genus 2 equations

Hilbert 10th problem, while undecidable in general, remains open for 2-variable equations: we do not know if there is an algorithm that, for polynomial $P(x,y)$ with integer coefficients, decides ...
Bogdan Grechuk's user avatar
11 votes
1 answer
526 views

Prove that $1$ is the sum of three tetrahedral numbers infinitely many different ways

It's well known that $1$ is the sum of three cubes infinitely many different ways but is it true for perhaps the tetrahedral numbers as well? Let $T_n = (1/6)n(n+1)(n+2)$. Then the following are the ...
Benjamin L. Warren's user avatar
4 votes
0 answers
235 views

Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?

When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...
Bogdan Grechuk's user avatar
2 votes
1 answer
354 views

Positive divisors of $P(x,n)=1+x+x^2+ \cdots + x^n$ that are congruent to $1$ modulo $x$

This is a follow-up question to Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$ Let \begin{equation} P(x,n)= 1+x+x^2+ \cdots + x^n, \end{equation} \begin{...
ASP's user avatar
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5 votes
0 answers
202 views

Integer points of rational function in 2 variables

Is there an algorithm that, given polynomials $P(x)$ and $Q(y)$ with integer coefficients, decides whether there exists integers $x$ and $y$ such that $\frac{P(x)}{Q(y)}$ is an integer? This is a ...
Bogdan Grechuk's user avatar

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