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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System of diophantine equations related to Ozanam's problem
Could you please help with finding of general solution of diophantine system for rational a, b, c, d
$(a^2+b^2)(c^2+d^2)=A^2$
$(a^2-b^2)(c^2-d^2)=B^2$
for some rational A and B.
This is related ...
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Simple Diophantine equations for Cartan matrices of Kac-Moody algebras
Let us consider the matrix $A$ defined as follows: $A_{i i} =2$,
$A_{i j} = - \frac{2 d_i}{d - 1 - d_{j}}, \qquad i \neq j$;
$i, j = 1, \dots, n$. Here $d_1,...,d_n$ are natural numbers, $n > 1$ ...
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A trivial application of Wilson's theorem to Brocard's Problem
Proposition: Let $W(1)$ be the set of all Wilson primes of order $1$ and suppose $n=p-1,$ where $p$ is a prime such that $p\notin W(1)$, then there are no integer solutions to the equation
$$n!+1=m^2$$...
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The existence of solution for special equation on integer ring
I have a question which belongs to the field of number theory. Can we prove or disprove the following claim:
For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one ...
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When is a cubic polynomial a cube? [closed]
I've been researching cubes and I'm trying to solve this Diophantine equation over the integers.
$$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = 5$...
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Question about link between non-terminals of grammars and variables of Diophantine equations
If we change the right arrow in the rewriting rules of grammar into equators , changes all terminals into x and keep the non-terminals unchanged,we get system of equations.In some cases,those ...
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Cassels-Birch-Davenport theorem for multiple quadratic forms of certain type
A classical theorem of Cassels states that if a homogenous quadratic form $Q$ has an integer zero, then there is a zero of small height (bounded solely by the coefficients and number of variables). ...
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Integral points on affine rational curves over $\mathbb{Q}$
Given a rational curve $C:(f_1(t),f_2(t))$, where $f_i(t),i=1,2$ are rational functions with rational coefficients.
Question: Is there any criterion(proved or conjectural) for the existence of ...
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Rational solutions of equations of the form $y^2 x = f(x)$
Let $k$ be any number field, and suppose we want to study the $k$-rational points on
$$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a ...
user40810
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Can six square numbers be simultaneously represented in a single sum of consecutive odd numbers? [closed]
I had some free time from my work to do a little exploration regarding the existence (or non existence) of perfect cuboids. A solution is represented by the set of Diophantine equations:
$a^2 + b^2 = ...
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Diophantine equation over Z[i]
I'm trying to generate the set of solutions of a specific diophantine equation over Z[i].
The equation is the following:
$$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$
with $ z_1$ (resp $z_2$) such that $\...
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Quartic case of a theorem of Bombieri and Pila
I am interested in the ternary case of a theorem of Bombieri and Pila, in E. Bombieri and J. Pila, "The number of integral points on arcs and ovals", Duke Mathematical Journal., 59 (1989), 337-357. ...
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If $n=x^k+y^k$ then also $n=a^2+b^2=c^3+d^3=\ldots =x^k+y^k$ [closed]
Are there infinitely many positive integers with the property:
If $n$ is a sum of two $k$th powers then it is also the sum of two $k-1$th powers, the sum of two $k-2$th powers, ... , the sum of two ...
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Solutions of system of diophantine equations
The system of diophantine equations $$\{x^2-y^2+z^2-u^2+q^2-t^2=0,\,xy+zt-uq=0 \}$$ is given. Do the formulas
$$x:=(j(p^2-4ps+3s^2)-(p-s)(3p^2-4ps+s^2))k^2+2(j-2(p-s))(p-s)kn+(j-p+s)n^2, $$
$$y:=(p-s)(...
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non negative integer solutions : Diophantine Equations [closed]
I want to know the exact number of non-negative integer solutions of $a_1 + 2a_2 + \ldots + k \cdot a_k = n$.
I know that it is the co-efficient of $x^n$ in $(1 - x^{a_1})^{-1} \cdot (1 - x^{a_2})^{-...
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On the equation $y^2 = x^3 - z^3$ [closed]
What is the parametric form of the rational solutions of the equation $y^2 = x^3 - z^3 ?$
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Classification of these Binary Quadratic Forms
What are necessary and sufficient conditions on a binary quadratic form $ax^2+bxy+cy^2$, with integer coefficients and solution set in integers, to be equivalent to $x^2-y^2$, and separately to $x^2+y^...
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On the elliptic curve $y^2 = x^3 + z^{4k}$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some $k \in \mathbb{Z}_{>1}$ ?
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$y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves
What are the rational solutions to the equation
$$
y^3 = x^4 + x,
$$
in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$?
Context: This is the simplest-looking ...
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Diophantine equations and modular forms
Let $D$ be a square-free positive integer which is the fundamental discriminant of a real quadratic field. Consider the following quadratic form $$Q_{D}(x,y)=x^2+Dy^2.$$
My questions are :
What is ...
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Special type Diophantine equations with integer solutions
The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method.
Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? ...
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The number of solutions of a Diophantine equation [closed]
Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...
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System of Diophantine equations
$p + p' = m$
$q - q' = n$
$pp' = qq'$
$(m^{2} + n^{2})\equiv1\pmod 4$ and $n^{2}\equiv0\pmod 4$.
Only $m,n$ are known in the above. Are there any known techniques to guess the values of $p$ and $q$...
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Solvability of two-variable quadratic equations with a parameter
(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation
$$
2 x^2+a x y+y^2+1 = 0
$$
is solvable in integers $(x,y)$.
(b) The same question for a similar equation
$...
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Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
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How can we solve the following number theory problem? [closed]
Let $m$ and $n$ be positive integers less than $2000$ which satisfies the equation $(m^2-mn-n^2)^2=1$. How can we determine the largest possible value of the expression $m^2+n^2$?
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Fastest way to solve non-negative linear diophantine equations
Let $A$ be a matrix in $M_{n \times m}(\mathbb{Z}_{\ge 0})$ without zero column. Let $V$ be a vector in $\mathbb{Z}_{> 0}^m$.
Question: What is the fastest way to find all the solutions $X \in \...
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Diophantine equations that involve cubes and the volume of square frustums
This week I wondered about diophantine problems that involve the volume of certain cubes and frustums, see the Wikipedia Frustum. I wondered if each one of these problems have infinitely many ...
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On the Diophantine equation $x^{4}+y^{4}= z^n$
I'm told that Ellenberg and Bruin proved that the Diophantine equation $x^{4} + y^{4} = z^n$ has no primitive solutions in which $xyz\neq 0$ and $n\geq 4$. Does the proof use the same methods that ...
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An asymptotic expression for the solution to the squares problem suggested by statistical mechanics
The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or ...
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Is surface $x^2+z^2=2\cdot y^2$ something of a Möbius strip?
This question is naive. My association with Möbius strip comes from not being able to smoothly extract positive solutions of the diophantine equation
$$x^2+z^2=2\cdot y^2$$
I got a parametrization (...
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The "universal" diophantine equation
There is a diophantine equation in some number (I think the minimum is now 9) of variables, that can be used to represent
All other diophantine equations (could be wrong on this)
Any particular set ...
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Diophantine equations involving recurrence sequences
I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
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Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?
Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
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Find the positive integers $x^3+y^3=3z^3$ [closed]
By Fermat Last theorem, I don't know if that's been discussed.
Find all positive integers $x,y,z$ such
$$x^3+y^3=3z^3$$
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Are there infinitely many solutions of $2^k=3^z-1$ with $k,z \in \mathbb{N}$? [duplicate]
This question arose as an attempt to answer the following question Relaxed Collatz 3x+1 conjecture. I wanted to show that there is a solution of the equation $2^{k}=3^{z}(2n+1)-1$ for each $n\geq 2$,
...
user111966
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Counting solutions of a certain diophantine equation
For some $s, k$, let $J_{s, k}(X; \mathbf{n})$ be the number of solutions to the system $$\sum_{i\le s} (x_i^j - y_i^j) = n_j$$ for $j\le k$ with $x_1, \dots, x_s, y_1, \dots, y_s\in [1, X]\cap\mathbb{...
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Find the rational cases where ${t}^{2} - 4$ is a perfect square with height bound $|t| \le N$ for positive integer $N \ge 1$
Find the unique cases when ${t}^{2} - 4$ is a perfect square say, ${n}^{2}$, with height bound $|t| \le N$ for positive integer $N \ge 1$, when $t$ is a rational where $t = p/q$ and integers $p$ an $q$...
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Quadratic diophantine equation in $\mathbb C[T]$
I am trying to solve the following quadratic diophantine equation in $\mathbb C[T]$, but I did not manage. I hope someone could give some hints or solutions to my problem. Here is the equation
$$\...
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Linear diophantine equation in n variables
Let n>3. Is there any way to generate all integer solutions of linear diophantine equation in n variables, or at least to determine number of such solutions?
Thanks in advance.
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Rational solutions of homogeneous equations
Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\...
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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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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 $p=x^2+3y^2$...
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Reference request: on sums of the form $ax^m + by^n = h$
I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
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Like Diophantine equation
Dear all,
I have posted this question on m.s.e. Unfortunately, no one responded to answer. I hope, this site and members of this site will answer my questions.
The equation $x^n - ny^x-nxy$ = $0$ ...
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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
$$ \|A q + p \| \ge c \| ...
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Diophantine equations that involve Lehmer means with all digits equal to $1$ in their $x-$adic expansions
In this post I present my variations of the problem involving Nagell-Ljunggren equation, that is explained in pages 10 and 11 of Highlights in the Research Work of T. N. Shorey by R. Tijdeman, from ...
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All the integer solutions of a certain semi-algebraic system
I would like to find all integer solutions of the following system:
$$a+b+c+ab+ac+bc=-2,$$
$$a,b,c\le a+b+c-1.$$
One solution is $2,2,-2$. Is it possible to describe all others?
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Solutions of the equation $X^4-DY^4=z^4$
In order to obtain infinite integer non trivial solutions of the equation $X^4-DY^4=Z^4$ (all numbers natural) we do the following.
We set $X=(r_1●p+p)$, $Y=p$, $Z=(r_1●p)$, $D=(4r_1^3+6r_1^2+4r_1+1)$...
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Bilinear system of Diophantine Equations
$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns and $n_{ij} \in \mathbb{Z}$ with $i \le j$ be the knowns.
Consider the following $\frac{n(n+1)}{2}$ with $n > 2$ ...
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