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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Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
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Beyond pure rational and integral solutions to cubic equations
I started reading Silverman and Tate’s introductory book on elliptic curves. In the introductory chapter they mention that for the Bachet equation $x^2 - y^3 = c$, there are infinitely many rational ...
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Solutions to diophantine equation related to an interpolation problem on hypercubes
Question:
which $n$ and $k$ satisfy $\frac{k^n-1}{2^n-1}\in\mathbb{N}$?
The motivation for the question is a constraint on the cardinality of interpolation-constraints for the $2^n$ corners of a ...
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Mod n, are all higher powers also lower powers?
Since there are only finitely many residues mod $n$, there is some function $H(n) \le n$ such that for all integers $n>1$, $r$, and $e>H(n)$, if $r$ is an $e$-th power mod $n$ then there is some ...
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Rational points on regular curves over global fields
Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...
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The security of one-time digital signatures from a solution to a diophantine equations
I wonder how well arbitrary Diophantine equations can be used to make one time digital signature schemes.
For our one-time digital signature scheme, the public key is a collection of polynomials $f_1(...
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About Diophantine equations
I am a self studying student and I am interesting with diophantine equations. I have the following questions:
How can I know that a diophantine equation is solved or not?
the second question is what ...
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Integers solutions of products of truncated Riemann zeta functions
Let $n \in \mathbb{N}$ be a positive integer.
It is possible to prove that the equation $F_1(n)=m$ where $m \in \mathbb{Z}$ and
$$
F_1(n)=(1+2+\ldots+n)\cdot\left(\frac{1}{1}+\frac{1}{2}+\dots+\frac{1}...
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Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
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Classifying solutions of a certain Diophantine Equation
The following question arose from a problem I am working on.
Let $N, k$ be positive integers. Consider the Diophantine equation in $a, b, c$:
$$
\frac{1}{a} + \frac{N - 1}{b} = \frac{N^k}{c}
$$
with ...
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2
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Integral solutions of quadratic equation $5 X² − 14 XY + 5 Y² = n$
Solve for all integers $x$ and $y$ the quadratic form $5 X² − 14 XY + 5 Y² = n$ for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these ...
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Can $P(z)$ have a divisor in a given congruence class?
In the answer to this previous question , Noam D. Elkies proved that for any integer $x$, $x^3-x^2-2x+1$ can only have a divisors equal to $-1$, $0$, or $1$ modulo $7$. I would like to know what is ...
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Can $z^3+1$ be powerful for integer $z$ other than $-1,0,2$ and $23$?
In 1976, Schinzel and Tijdeman proved that if a polynomial $P(z)$ with integer coefficients has at least $3$ simple zeros, then there may be at most finitely many $z$ such that $P(z)$ is a perfect ...
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How to describe all integer solutions to $x^2+y^2=3z^2+1$?
The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is
$$
x^TAx+bx+c=0,
$$
where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with ...
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On GCD and lattice reduction
$LLL$ algorithm is vectorized version of Euclidean algorithm for $GCD$.
Even the $m=2$ case known to Lagrange and Gauss does not have an $NC$ algorithm for shortest vector.
If $GCD$ is in $NC$ and in ...
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Rational solutions to Catalan's equation
Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation
$$
x^{a}-y^{b}=1.
$$
for $a, b > 1$ and $x, y > 0$ is $x = 3,...
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Trigonometric Diophantine equation
Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number?
This ...
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Small Galois group solution to Fermat quintic
I have been looking into the Fermat quintic equation $a^5+b^5+c^5+d^5=0$. To exclude the trivial cases (e.g. $c=-a,d=-b$), I will take $a+b+c+d$ to be nonzero for the rest of the question. It can be ...
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Size of sets associated to Gaussian integers
Given a non-zero Gaussian integer $z$, we define the set $\mathcal S(z)$
containing all solutions
of $ab+cd=z$ satisfying $\min(\vert a\vert,\vert b\vert)>\max(\vert c\vert,\vert d\vert)$ with $a,b,...
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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. ...
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On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two
I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) ...
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How to find rational points on genus 2 rank 2 curves such as $y^2=x^6-4x+4$?
The question is in the title. The motivation comes from trying solving Diophantine equations in order, see Can you solve the listed smallest open Diophantine equations? . Because there is an algorithm ...
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Simple motivation to study arithmetic geometry
Is there a simple-to-understand diophantine equation (in the sense that it's easy to explain to a child) that has a positive integer solution, but to prove that such a solution exists and to find it ...
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How constructive is Matiyasevich's theorem?
A famous corollary of Matiyasevich's theorem is that there exists a Diophantine equation such that it is undecidable (under some recursively axiomatizable theory $T$, such as ZFC) whether that ...
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Only trivial solution to a pair of constrained linear diophantine equations
Given positive integer $n$, we are looking for a set
of $n$ positive integers $a_i$.
The following linear integer program must have only
the trivial integer solution of all ones.
$0 \le x_i \le \frac{...
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2
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Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
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What numbers have three Pythagorean partners in geometric progression?
Call two positive integers $\{m, n\}$ "Pythagorean partners" (PPs) if the sum of their squares is a square. Since all such pairs are of the form $\{2rpq, r(p^2-q^2)\}$ for some positive ...
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Reference for bound $x_{0} \le n^{2}$ in solutions to the Diophantine equation $\left(\sum_{i=1}^{n} x_{i}\right)^{2} = x_{0}\prod_{i=1}^{n}x_{i}$?
Using Vieta jumping, one can prove that, if $x_{0}, x_{1}, \dotsc,
x_{n} \in \mathbb{Z}_{\ge 1}$ are such that
\begin{equation*}
\left(\sum_{i=1}^{n} x_{i}\right)^{2} =
x_{0}\prod_{i=1}^{n}x_{i},...
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The "Smallest" open Diophantine Equation, a potential approach
In this question, Bogdan Grechuk found the "Smallest" open Diophantine equation, where size is determined by taking absolute values of all coefficients, then substituting 2 into all ...
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A diophantine equation inspired in a conjecture due to Gica and Luca, example of a large Mersenne exponent
In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$
over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and very large Mersenne exponents $x$ such that $x^2-2$ ...
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Polynomial parametrization for solutions of quadratic Diophantine equations
A previous Mathoverflow question asks if there is an algorithm that would determine all integer solutions to a given quadratic Diophantine equation.
To make this question more formal, we need to agree ...
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Reverse engineering a Diophantine equation
Recently, due to the help I had with another question, I was able to find a Diophantine equation of degree in four variables which is the condition to be able to construct a "rational" ...
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Bounds on Bézout coefficients
Let $0<a_1 \le a_2 \le \cdots \le a_n$ be positive integers such that $a_1 + \cdots + a_n = m$ and $\gcd(a_1,\ldots,a_n)=1$. Let $\mathbf a :=(a_1,\ldots,a_n)\in\mathbb Z^n$ and $\mathbf x:=(x_1,\...
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2
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Solutions to a system of Diophantine equations
In my research in a different field (representation theory), the following system of equations popped up:
$$
ax=by
$$
$$
xy+a+b-ax=p
$$
where $p\in\{0,1,2,3,4\}$ and $a,b,x,y$ are integers (I am also ...
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Formula for "cointersection" of three circles?
I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?
...
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Fastest way to solve non-negative linear diophantine equations
Let $A$ be a matrix in $M_{n \times m}(\mathbb{Z}_{\ge 0})$ without zero column. Let $V$ be a vector in $\mathbb{Z}_{> 0}^m$.
Question: What is the fastest way to find all the solutions $X \in \...
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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 ...
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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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2
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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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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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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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1
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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
$$
U_S = \{t' \in \mathbb{...
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Is equation $xy(x+y)=7z^2+1$ solvable in integers?
Do there exist integers $x,y,z$ such that
$$
xy(x+y)=7z^2 + 1 ?
$$
The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we ...
1
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0
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Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?
I posted this question on SE, and was told I should repost it here.
The Goormaghtigh conjecture explores the Diophantine equation of the form
$$
\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1},
$$
where $a>c&...
8
votes
1
answer
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Hilbert 10th problem for cubic equations
Hilbert 10th problem, asking for algorithm for determining whether a polynomial Diopantine equation has an integer solution, is undecidable in general, but decidable or open in some restricted ...
3
votes
1
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343
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Rational points of bounded height on a variety
I would like to ask for some clarification on the following argument which I can not quite understand.
There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
0
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0
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Integral solutions to f(x, y, z) = n where f is a cubic form
I'm looking to see if there is an integral solution to $f(x,y,z)=n$ where f is a cubic form. Especially interesting is the diagonal case:
$$
ax^3+by^3+cz^3=n
$$
for fixed integers $a,b,c,n$. If there ...
0
votes
2
answers
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The (last step of the) proof that the set of badly approximable matrices has measure zero
An $m \times n$ matrix $A$ is called badly approximable if there exists $c > 0$ such that for all integer vectors $p \in \mathbb Z^m$ and $q \in \mathbb Z^n-\{0\}$ we have
$$ \|A q + p \| \ge c \| ...
1
vote
1
answer
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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 $p=x^2+3y^2$...