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Results tagged with nt.number-theory
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user 89064
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
6
votes
0
answers
105
views
Existence of large integer solution for a simple-looking equation
Is it true that for every $k>0$ Diophantine equation
$$
y^2 + x^2y + z^2x + 1 = 0
$$
has an integer solution $(x,y,z)$ such that $\min\{|x|,|y|,|z|\}\geq k$?
Motivation: this equation arises in the st …
- 4,905
22
votes
1
answer
984
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 coe …
- 4,905
5
votes
2
answers
396
views
$x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations
(a) Do there exist integers $x$ and $y$ such that $x^3+x^2y^2+y^3=7$ ?
(b) Is this equation belongs to some family $F$ of equations for which there is a known algorithms for testing if they have an i …
- 4,905
6
votes
5
answers
693
views
What are the integer solutions to $z^2-y^2z+x^3=0$?
The question is to describe ALL integer solutions to the equation in the title. Of course, polynomial parametrization of all solutions would be ideal, but answers in many other formats are possible. F …
- 4,905
0
votes
0
answers
149
views
Representing integers as sums of three powers
A famous open question, discussed several times on MathOverFlow, asks Which integers can be expressed as a sum of three cubes in infinitely many ways?. This is open even for $n=3$, that is, we do not …
- 4,905
9
votes
1
answer
686
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 $y^2-4$?
T …
- 4,905
10
votes
3
answers
680
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 …
- 4,905
10
votes
1
answer
319
views
Positive integers such that $(x+y)(xy-1)=z^2+1$
Do there exist positive integers $x,y,z$ such that
$$
(x+y)(xy-1)=z^2+1
$$
In my previous question Can you solve the listed smallest open Diophantine equations?, I discuss the smallest equations for w …
- 4,905
5
votes
1
answer
306
views
Are these equations solvable in positive integers?
By Matiyasevich theorem, there is no algorithm to decide whether a given Diophantine equation $P(x_1,\dots, x_n)=0$ has a solution in positive integers. As suggested in What is the smallest unsolved D …
- 4,905
2
votes
0
answers
35
views
Are there integers $x,y,z$ such that $1 + x - x^3 + x^2 y^2 + z + z^2 = 0$?
In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any inte …
- 4,905
5
votes
0
answers
200
views
Are there integers $x,y,z$ such that $(x+1)y^2-xz^2=x^3+2x+2$?
Is equation
$$
(x+1)y^2-xz^2=x^3+2x+2
$$
solvable in integers?
Motivation: For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, define …
- 4,905
8
votes
1
answer
495
views
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 familie …
- 4,905
5
votes
0
answers
326
views
On the shortest open cubic equation
The question is: are there any integers $x,y,z$ such that
$$
1+4 x^3+x y^2+2 y z^2 = 0 \quad\quad\quad\quad (1)
$$
The motivation is: Define the length of a polynomial $P$ consisting of $k$ monomials …
- 4,905
65
votes
1
answer
6k
views
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 e …
- 4,905
1
vote
0
answers
424
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 integ …
- 4,905