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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On the equation $a^4+b^4+c^4=2d^4$ in natural numbers with $a<b<c<d$
I asked a simillar question with the weaker restriction:
On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$
.
I couldn't find any solution to this equation. ...
user178594
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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 ...
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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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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 $|w||z|-xy=1$
$$w,z<0<\max(...
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On the equation $x^3 + y^3 =cz^3$
What are the characteristics of the values of $c$ for which the equation $x^3 + y^3 = cz^3$ has pairwise coprime non-zero integral solutions where $x \neq \pm y$ ? For instance, it is known that $c$ ...
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Polynomial solutions of equation $P(t) - T_n(x)$ with Chebycheff polynomials
Suppose we have, for fixed $P(t) \in \mathbb{Z}[t]$, infinite couples of integer solutions $(t_i, x_i) \in \mathbb{Z}$ to the equation
$$ P(t) - T_n(x) = 0 $$
where $T_n$ is the $n$-th Chebycheff ...
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Sign Enumeration
What is the number of solutions of $(a_i)_{i=1}^n$ such that
$$\sum_{i=1}^nia_i\le b,\quad a_i\in\{-1,1\},\quad \sum_{i=1}^n{a_i}=c$$
given $b,c\in\mathbf Z$?
Is there a generating function solution?
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On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk
I. Elliptic curves
Given integers $a,b,m_k$. Let,
$$x^2+a = m_1u_1^2\\x^2+b = m_1u_2^2\tag1$$
If there is a rational point $x_i$, then the pair (after a transformation) is birationally equivalent ...
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Non-coprime solutions to x^n+y^n = z^2
Let $n$ be an odd prime. I know that the equation $x^n+y^n = z^2$ has no non-zero coprime solution in integers whenever $n \geq 5$, and that there are infinitely many solutions as soon as one drops ...
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Help with this Diophantine equation
Note: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at Help with this system of Diophantine equations (End of note)
For a research ...
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A problem on cubic Diophatine equations
What is the best algorithm to find all the integer points (X,Y) on this curve
$X^3+aX-bY^3=m,a,b,m\in\mathbb{Z}$(a>0,b>0,b is not a cubic number)?
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Diophantine equation problem
How many positive integer solutions does the equation x^2+y^2+z^2-xz-yz = 1 have? My guess is (1,0,1), (0,1,1) and (1,1,1). What is proof of that fact that there are none other?
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How to prove that this equation has no other integer solutions?
To find the integer solutions of an indeterminate equation and prove that there are no other solutions, where all variables are positive integers and n can be regarded as a constant, let's first ...
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Solutions to a system of Diophantine equations
In my research in a different field (representation theory), the following system of equations popped up:
$$
ax=by
$$
$$
xy+a+b-ax=p
$$
where $p\in\{0,1,2,3,4\}$ and $a,b,x,y$ are integers (I am also ...
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generalizations of the delone nagell equation
Are there any references that study integer solutions to cubic Diophantine equations of the form $x^3 + 2y^3 = 2^a 3^b$ for $\{a,b\}\subset \{0,1,2,3\}$? I am aware that Nagell solved $x^3 + 2y^3 = 1$ ...
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Diophantine equations that involve Gregory coefficients: a computational exercise
In this post that I've asked three weeks ago with same title in Mathematics Stack Exchange and identificator 3692235, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has ...
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Are there any nonzero rational solutions to this equation?
Are there any nonzero rational solutions to the equation
$$y^2 = 64x^n + 1$$ where $n\geq 3$ is an integer ?
For the case $n=3$, the question can be settled by basic ideas of elliptic curves, but i'...
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how to solve this symmetrical equation in number theory
i just have no idea about this equation, i would thank you to you to give me some suggestions on this.
$$m_{1}m_{2}m_{3}+2^{\alpha-s-t}m_{1}+2^{\alpha-\gamma-t}m_{2}+2^{\alpha-\gamma -s}m_{3}\\-2^{\...
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basis of the lattice generated by the integer points inside a subspace of R^L
Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...
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The existence of solutions of a Diophantine exponential equation
Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation
$$p^x+p^n=\sum_{i=0}...
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A Mordell equation $y^3=x^2+20$ [closed]
Recently I met a problem when I'm studying algebraic number theory.
Problem. Find all positive integer solutions of $y^3=x^2+20$.
I solved the situation when $x$ is an odd because the two ideals $(x+...
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Solutions to diophantine equation related to an interpolation problem on hypercubes
Question:
which $n$ and $k$ satisfy $\frac{k^n-1}{2^n-1}\in\mathbb{N}$?
The motivation for the question is a constraint on the cardinality of interpolation-constraints for the $2^n$ corners of a ...
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Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
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Diophantine equations with arithmetical functions
I want to know is the diophantine equations that contain arithmetic functions are an interesting topic to research? (For example $\varphi(x)=cx-1$ and $\varphi(x)=\sigma(x)-1$.)
$\sigma(x)$ is the sum ...
user482376
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On a quadratic diophantine equation
Given a quadratic diophantine equation in $\mathbb Z[x,y,z]$ of form
$$ax^2+by^2+cx+dy+ez+f=0$$ are there standard methods to solve for it when $$\|(x^2,y^2,z)\|_\infty\leq e^{1/2}$$
$$\|(a,b,c,d,e,f)...
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Dimension of $S$-units over $\mathbb{C}[x]$
Let $S=\{s_1,\ldots,s_n\}$ be a finite set of complex numbers. Consider the set of polynomials $$U=\{\,(x+s_1)^{k_1}\cdots(x+s_n)^{k_n}:\, 0\leq k_1+\cdots+k_n\leq H\}.$$
I am curious as to what is ...
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Question in the setting of generalized Diophantine $m$-tuples
As an amateur I am not quite sure should I post a question on the site for professional mathematicians but if the question is not appropriate for this site you can freely migrate it to ...
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Is there (conjectural) upper bound for the largest solution of Diophantine equation with finitely many solutions?
Let $F(x_1,\ldots,x_n)$ be polynomial with integer coefficients
and $x_i$ integers.
System of Diophantine equations can be brought in this form
via sum of squares.
Assume $F=0$ has finitely many ...
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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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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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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 ...
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Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application
Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil<
r_1,r_2<\lceil1+\sqrt{p}\rceil$ and
$$r_1\equiv mac\bmod p$$
$$r_2\equiv mbd\bmod p$$...
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A paper by W. Ljunggren
I am looking for the following paper by Ljunggren, Wilhelm: "Zur Theorie der Gleichung $x^2 + 1 = Dy^4$", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27
The main result of this paper which I am ...
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A question about integer triples
How can we generate all integer solutions of the equation
$$(qr+rp+pq)(x^2+y^2+z^2) = (p^2+q^2+r^2)(yz+zx+xy),$$
given that $p,q,r$ are integers?
Clearly if any one of $(x,y,z), (x,z,y), (y,z,x), (...
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Upper bound for a subset of $\mathbb{N}^2$
Question: Consider the set
$$ A(m) : = \{ (a, b) \in \mathbb{N}^2 : \; N \leq a \leq 2 N, \; a^2 - b^2 = m \}, $$
where $m \in \mathbb{Z}$ and $N \in \mathbb{N} \setminus \{ 0 \}$. Then
$$ \...
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Solutions to Diophantine equation for Ramanujan graph construction
I am working with so-called Ramanujan graphs which have the property that a lower bound on their girth can be stated. I am reading about those in paper On the Construction of Turbo Code Interleavers ...
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Derangements and unit fractions
Motivated by a recent question of Zhi-Wei Sun and its nice answer by Zhao Shen, here are two related questions.
Let $S_n$ be the group of permutations on $\{1, 2, \ldots, n\}$.
a. For each $n \ge ...
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Sharply Estimating Pythagorean Triples [closed]
Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?
user76479
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How to solve a quadratic diophantine equation [closed]
I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the ...
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Paired Quadratic diophantine equations
For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$
If so, how to ...
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System of linear diophantine equations with many small solutions?
Let $n$ be positive integer, $k$,$B$ fixed positive integers.
Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.
Let $S(f_i,k,B)$ be the set of ...
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Eisenstein triples (and triangles with rational sides and a rational-degree angle) in Pascal's triangle
This question leads to a follow-up: are there any Eisenstein triples (satisfying $a^2\pm ab+b^2=c^2$) in one row of Pascal's triangle apart from the following:
$\binom{23}{8}^2+\binom{23}{8}\binom{23}{...
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Infinite number of decompositions into sum of four cubes
The context is the sum-of-four-cubes problem (see here).
I ask myself the following question (I asked a similar question on MSE). Do we know if any integer $k=a^3+b^3+c^3+d^3$ can be decomposed in an ...
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Interesting solutions of equation x^y = y^x [closed]
There is simple equation $x^y=y^x$. By taking logarithm we can see that it is equivalent to $\frac{\ln x}{x}=\frac{\ln y}{y}$. When we plot and inspect the function $f(x)=\frac{\ln x}{x}$, we can see ...
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Integral points on "complex exponential surface" in $\mathbb{C}^3$
I encountered the following object in $\mathbb{C}^3$ defined for $m\in\mathbb{N}$ by
$$A_m:=\lbrace (z_1,z_2,z_3)\in\mathbb{C}^3|(2^{2z_3}m-1)2^{2z_1+z_2+1}+3^{z_2-1}(2^{2z_1}-2^2-3^{z_3+1}m)=0\rbrace$...
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Situations where the number of solutions to a linear Diophantine equation is always even
I have a number theory situation that I hope someone will recognize as a known situation and can direct me to some relevant papers in the literature.
This came out of some numerical experiments run by ...
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On the equation $q(\mathbf{x}) = 1$ for $q$ a quadratic form
Let $q(\mathbf{x}) = q(x_1, \cdots, x_n)$ be a quadratic form with integer coefficients. For $n \geq 3$, is there a reasonable theory for the set of integer solutions to the equation
$$\displaystyle q(...
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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 ...
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Diophantine equation Oeis A159589
Considera the diophantine equation:
$y^2=x^2+(x+449)^2$.
Is there a method to solve this equation?
And why an Oeis sequence Is dedicated to this equation? Has this diophantine equation something ...
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Mahler's proof of $S$-unit equation
Many modern proofs of the (ineffective) finiteness of solutions of the $S$-unit equation $x+y=1$ use Roth's theorem. In particular it is used Lang's version of Roth's theorem which takes in account ...
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