All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
1
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Diophantine equations with arithmetical functions
I want to know is the diophantine equations that contain arithmetic functions are an interesting topic to research? (For example $\varphi(x)=cx-1$ and $\varphi(x)=\sigma(x)-1$.)
$\sigma(x)$ is the sum ...
user482376
3
votes
1
answer
279
views
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 ...
user482376
2
votes
2
answers
1k
views
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
$3x^2+3y^2+z^2=t^2$
...
- 21
4
votes
2
answers
574
views
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 ...
- 91
7
votes
2
answers
398
views
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 $z^n - z$...
- 1,625
2
votes
1
answer
259
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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
$$
U_S = \{t' \in \mathbb{...
- 8,998
30
votes
2
answers
1k
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Is equation $xy(x+y)=7z^2+1$ solvable in integers?
Do there exist integers $x,y,z$ such that
$$
xy(x+y)=7z^2 + 1 ?
$$
The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we ...
- 7,182
1
vote
0
answers
243
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Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?
I posted this question on SE, and was told I should repost it here.
The Goormaghtigh conjecture explores the Diophantine equation of the form
$$
\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1},
$$
where $a>c&...
- 119
8
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1
answer
627
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Hilbert 10th problem for cubic equations
Hilbert 10th problem, asking for algorithm for determining whether a polynomial Diopantine equation has an integer solution, is undecidable in general, but decidable or open in some restricted ...
- 7,182
3
votes
1
answer
397
views
Rational points of bounded height on a variety
I would like to ask for some clarification on the following argument which I can not quite understand.
There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
- 8,998
0
votes
0
answers
126
views
Integral solutions to f(x, y, z) = n where f is a cubic form
I'm looking to see if there is an integral solution to $f(x,y,z)=n$ where f is a cubic form. Especially interesting is the diagonal case:
$$
ax^3+by^3+cz^3=n
$$
for fixed integers $a,b,c,n$. If there ...
- 9,114
0
votes
1
answer
419
views
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$...
- 7,182
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votes
2
answers
751
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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 ...
- 3,098
1
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0
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274
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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 ...
- 547
1
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0
answers
143
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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 ...
- 15.6k
8
votes
1
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729
views
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$?
...
- 7,182
1
vote
1
answer
192
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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 ...
- 13.9k
4
votes
0
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465
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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 ...
- 57
2
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1
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269
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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(...
- 15.6k
1
vote
0
answers
453
views
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. ...
- 7,182
2
votes
0
answers
242
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... ...
- 4,862
2
votes
0
answers
115
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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
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1
answer
180
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 ...
- 25.4k
9
votes
0
answers
274
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 ...
- 7,182
-4
votes
1
answer
200
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
$$ x^k+y^k-z^k-w^k=3\quad \ (x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})\tag{1}$$
...
- 15.6k
3
votes
1
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218
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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 ...
- 15.6k
21
votes
1
answer
739
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 ...
- 15.6k
4
votes
1
answer
306
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\...
- 15.6k
3
votes
0
answers
393
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
$x^3 - 1 = (x - k) y^3$,...
- 827
11
votes
0
answers
363
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+...
- 15.6k
0
votes
0
answers
535
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 ...
- 7,182
9
votes
3
answers
757
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 ...
- 7,182
5
votes
0
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284
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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
$$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
- 15.6k
2
votes
1
answer
278
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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 ...
16
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2
answers
857
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 $$(...
- 6,825
6
votes
1
answer
1k
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Find all integer solutions to the following easy-looking Diophantine equations
In general, it is not clear What does it mean to solve an equation? in integers. In this question, let us assume that an equation
$$
P(x_1,\dots,x_n)=0
$$
is solved if we have proved that its integer ...
- 7,182
0
votes
1
answer
125
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Special type of normal form of matrix in principal ideal domain
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric.
Can one always ...
- 145
1
vote
1
answer
274
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Prove there are infinitely many squares which are the sum of two tetrahedral numbers [closed]
Let $T_n = \frac{1}{6}n(n+1)(n+2)$ denote the $n$th Tetrahedral number. The first several solutions to squares as sums of two Tetrahedral numbers are {T_n,T_m,a^2}
1 5 6\
1 8 11\
1 22 45\
1 24 51\
1 ...
- 1,109
5
votes
6
answers
3k
views
How many cubes are the sum of three positive cubes?
Are there infinitely many integer positive cubes $x^3 = a^3 + b^3 + c^3$ that are equal to the sum of three integer positive cubes? If not, how many of them are there?
- 67
2
votes
3
answers
568
views
Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$
If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the equation
$$
yz=...
- 7,182
1
vote
1
answer
144
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 $|w||z|-xy=1$
$$w,z<0<\max(...
- 13.9k
24
votes
2
answers
1k
views
On the smallest open Diophantine equations: beyond Hilbert's 10 problem
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all ...
- 7,182
3
votes
0
answers
333
views
What does it mean to solve an equation?
Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation
$$
P(x_1,\dots,x_n) = 0
$$
where $P$ is a polynomial with integer coefficients. Do we have a ...
- 7,182
16
votes
2
answers
1k
views
Representing $x^3-2$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.
Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
- 7,182
7
votes
0
answers
237
views
Magic hourglass of squares hyperelliptic equation
I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so:
$a^2$ $b^2$ $c^2$
$ $ $ $ $ $ $ $ $ $ $d^2$
$e^2$ $f^2$ $g^2$
...
- 2,811
6
votes
1
answer
438
views
$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
- 7,182
1
vote
0
answers
261
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 ...
- 7,182
1
vote
0
answers
93
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 ...
- 145
5
votes
0
answers
333
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 ...
- 159
1
vote
0
answers
91
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 \\
\...
- 149