# 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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### 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 $3x^2+3y^2+z^2=t^2$ ...
1 vote
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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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### 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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### 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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### 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 ...
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 ...
1 vote
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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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### 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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### 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 ...
1 vote
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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)...
1 vote
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### 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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### $y^3=x^4+x+1$, 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $P(x)$, ...