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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Diophantine equations

It has been proved that there is no algorithm to solve Diophantine equations, for that reason I want to know what are the Diophantine equations that physicists or chemists need to solve? Or any other ...
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24 votes
4 answers
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Why does representing functors help solving Diophantine equations?

Here I read: Another insight of Grothendieck and his school was, how important it is to represent functors in algebraic geometry - regardless of what you want at the end. [as Mazur reports, Hendrik ...
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Parametrization of integral solutions of $3x^2+3y^2+z^2=t^2$ and rational solutions of $3a^2+3b^2-c^2=-1$

1/ Is it known the parameterisation over $\mathbb{Q}^3$ of the solutions of $3a^2+3b^2-c^2=-1$ 2/ Is it known the parameterisation over $\mathbb{Z}^4$ of the solutions of $3x^2+3y^2+z^2=t^2$ ...
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Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms

I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations. Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
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7 votes
2 answers
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Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer?

I came across the following conjecture. If you have any thoughts on how to approach it, let me know. Conjecture. For any integer $n > 3$ and any Gaussian integer $z$ that is not a unit, if $z^n - z$...
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Rational points on a special class of surfaces

Consider a smooth surface of the following form $$ S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3 $$ over $\mathbb{Q}$, and set $$ U_S = \{t' \in \mathbb{...
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30 votes
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Is equation $xy(x+y)=7z^2+1$ solvable in integers?

Do there exist integers $x,y,z$ such that $$ xy(x+y)=7z^2 + 1 ? $$ The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we ...
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1 vote
0 answers
170 views

Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?

I posted this question on SE, and was told I should repost it here. The Goormaghtigh conjecture explores the Diophantine equation of the form $$ \frac{a^b-1}{a-1}=\frac{c^d-1}{c-1}, $$ where $a>c&...
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Hilbert 10th problem for cubic equations

Hilbert 10th problem, asking for algorithm for determining whether a polynomial Diopantine equation has an integer solution, is undecidable in general, but decidable or open in some restricted ...
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4 votes
1 answer
240 views

Rational points of bounded height on a variety

I would like to ask for some clarification on the following argument which I can not quite understand. There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
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Integral solutions to f(x, y, z) = n where f is a cubic form

I'm looking to see if there is an integral solution to $f(x,y,z)=n$ where f is a cubic form. Especially interesting is the diagonal case: $$ ax^3+by^3+cz^3=n $$ for fixed integers $a,b,c,n$. If there ...
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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 \| ...
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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 $p=x^2+3y^2$...
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7 votes
2 answers
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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 ...
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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 ...
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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 ...
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On the set $\{\delta+x^2+2y^2+3z^2+xyz:\ \delta=0,1;\ x,y,z=0,1,2,\ldots\}$

Let $\mathbb N=\{0,1,2,\ldots\}$. It is well known that any positive odd number can be written as $x^2+2y^2+3z^2$ with $x,y,z\in\mathbb N$. Thus $$\{\delta+x^2+2y^2+3z^2:\ \delta\in\{0,1\}\ \ \mbox{...
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8 votes
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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 $y^2-4$? ...
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1 vote
1 answer
101 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 ...
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4 votes
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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 ...
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2 votes
1 answer
238 views

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(...
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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. ...
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2 votes
0 answers
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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... ...
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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 ...
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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)...
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1 vote
1 answer
115 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 ...
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9 votes
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$y^3=x^4+x+1$, 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 ...
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-4 votes
1 answer
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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 $$ x^k+y^k-z^k-w^k=3\quad \ (x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})\tag{1}$$ ...
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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 ...
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14 votes
1 answer
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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 ...
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3 votes
1 answer
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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\...
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0 votes
0 answers
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Number of integer solutions of a certain polynomial system of equations

Let the homogeneous polynomials $f_1,f_2,f_3\in\mathbb{Z}[x_1,x_2,x_3]$ be defined by \begin{align}f_1&=x_1^2x_2^2+x_1^2x_3^2+x_2^2x_3^2, \\\ f_2&=x_1x_2x_3(x_1+x_2+x_3), \\\ f_3&=(x_1-x_2)...
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2 votes
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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 $x^3 - 1 = (x - k) y^3$,...
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4 votes
1 answer
241 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 ...
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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+...
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1 vote
0 answers
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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 ...
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9 votes
3 answers
606 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 ...
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5 votes
0 answers
272 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 $$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
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2 votes
1 answer
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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 ...
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15 votes
2 answers
644 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 $$(...
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2 votes
1 answer
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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 ...
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1 vote
1 answer
106 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 ...
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4 votes
1 answer
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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 ...
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1 answer
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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 ...
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4 votes
5 answers
2k 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?
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3 votes
3 answers
454 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=...
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1 vote
1 answer
125 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(...
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19 votes
1 answer
765 views

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 ...
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3 votes
0 answers
232 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 ...
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15 votes
1 answer
782 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 $P(x)$, ...
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