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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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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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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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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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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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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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 $$(...
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Find all integer solutions to the following easy-looking Diophantine equations
In general, it is not clear What does it mean to solve an equation? in integers. In this question, let us assume that an equation
$$
P(x_1,\dots,x_n)=0
$$
is solved if we have proved that its integer ...
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Special type of normal form of matrix in principal ideal domain
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric.
Can one always ...
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Prove there are infinitely many squares which are the sum of two tetrahedral numbers [closed]
Let $T_n = \frac{1}{6}n(n+1)(n+2)$ denote the $n$th Tetrahedral number. The first several solutions to squares as sums of two Tetrahedral numbers are {T_n,T_m,a^2}
1 5 6\
1 8 11\
1 22 45\
1 24 51\
1 ...
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How many cubes are the sum of three positive cubes?
Are there infinitely many integer positive cubes $x^3 = a^3 + b^3 + c^3$ that are equal to the sum of three integer positive cubes? If not, how many of them are there?
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Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$
If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the equation
$$
yz=...
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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 $|w||z|-xy=1$
$$w,z<0<\max(...
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On the smallest open Diophantine equations: beyond Hilbert's 10 problem
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all ...
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What does it mean to solve an equation?
Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation
$$
P(x_1,\dots,x_n) = 0
$$
where $P$ is a polynomial with integer coefficients. Do we have a ...
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answers
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Representing $x^3-2$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.
Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
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Magic hourglass of squares hyperelliptic equation
I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so:
$a^2$ $b^2$ $c^2$
$ $ $ $ $ $ $ $ $ $ $d^2$
$e^2$ $f^2$ $g^2$
...
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The strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation
Let us consider the strong twin conjecture:
For all positive integer $n$ there exist a prime $p$ such that
$$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime
Since the inequalities and the ...
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$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
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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.$ ...
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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 ...
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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 ...
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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 ...
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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 \\
\...
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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Diophantine equations that involve Lehmer means with all digits equal to $1$ in their $x-$adic expansions
In this post I present my variations of the problem involving Nagell-Ljunggren equation, that is explained in pages 10 and 11 of Highlights in the Research Work of T. N. Shorey by R. Tijdeman, from ...
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Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer.
While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...
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A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem
I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
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Can $2^n\pm n$ with $n>2$ be a triangular number?
Recall that triangular numbers are those
$$T(n)=\frac{n(n+1)}2\ \ (n=0,1,2,\ldots\}.$$
Clearly, $$2^1-1=1=T(1),\ \ 2^1+1=3=T(2),\ \ 2^2+2=6=T(3).$$
Question. Is there an integer $n>2$ with $2^n-n$ ...
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Software for detecting Brauer-Manin obstructions?
In the context of another MO question, the following question arose: Does there exist any software for detecting Brauer–Manin obstructions to the existence of integer solutions to a single polynomial ...
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How can I find all integer solutions of $3^n - x^2 = 11$
I know that $n$ can't be even because of the following argument:
Let $n = 2p$. Then we can use the difference of two squares and it becomes like this :
$(3^p + x)(3^p - x) = 11; 3^p + x = 11 , 3^p - x ...
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List of obscure summation identities [closed]
I am trying to evaluate a fairly simple summation:
$\sum_{k=1}^n ka^kb^{n-k}$
Which is related to the common identity for $\sum_{k=1}^n ka^k$ available on Wikipedia.
I've previously seen lengthy lists ...
21
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answer
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Is "almost-solvability" of Diophantine equations decidable?
Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the ...
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A family of Diophantine equations with no integer solutions but solutions modulo every integer
Selmer's curve is the equation $3x^3 +4y^3 +5z^3=0$. This equation is famous for having non-trivial solutions in every completion of $\mathbb{Q}$ but only having the trivial solution in the rationals. ...
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votes
3
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Can you solve the listed smallest open Diophantine equations?
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
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3
votes
1
answer
262
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Determine if a 2-variable Diophantine equation has a finite or infinite number of solutions
Do there exist an algorithm, which, given a polynomial $P(x,y)$ with integer coefficients, determines whether Diophantine equation $P(x,y)=0$ has finite or infinite number of integer solutions?
Famous ...
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A specific Diophantine equation related to the congruent number question
Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
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Laurent polynomials of the form $p(x)\cdot p(x^{-1})$
Let $R$ be a commutative, associative ring with $1$ and let $\tau: R[x^{\pm 1}]\to R[x^{\pm 1}]$ be the $R$-algebra involution $\tau(x)=x^{-1}.$ Then it is natural to ask which elements of the ...
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On the Diophantine equation $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$
This question is an offshoot of this closely related MO question.
Here, we consider the Diophantine equation
$$m^2 - p^k = 2^r t,$$
where $r \geq 2$ and $\gcd(2,t)=1$.
Furthermore, we place the ...
6
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1
answer
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views
Given some recursive function, can we effectively associate it a polynomial as in the DPRM theorem?
I'm interested in the following assertion about the Davis-Putnam-Robinson-Matijasevich theorem
Given a recursive function $f:\mathbb{N}\rightarrow\mathbb{N}$, i.e. its index, we can effectively get ...
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3
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Non-trivial solution to $\sum^{n}_{i=1}\sum^{n}_{j=1,j\ne i}(x_{i})^{(x_j)}=(\sum^{n}_{i=1}x_i)^{(\sum^{n}_{i=1}x_i)}$
This problem was first asked at Mathematics Stack Exchange, where it wasn't drawn much attention.
For ease of reading,
$$S=\sum_{i=1}^nx_i, S_p=\sum_{i=1,i\ne p}^nx_i, S^{[q]}=\sum_{i=1}^nx_i^q, S_p^{[...
9
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0
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Iterating Diophantine equations over Q to quickly get a large interval with just integer solutions
Hilbert's Tenth Problem was whether there is an algorithm which will answer whether any Diophantine equation has solutions (where we want integer solutions). Hilbert's Tenth has a negative solution by ...
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