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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Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
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Almost Pell type equation
Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
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Special type of normal form of matrix in principal ideal domain
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric.
Can one always ...
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find all of rational solutions of the quartic equation?
Consider the equation $a^4+6v^2a^2-8a+v^4=0$ over the rationals.
Note that the following are solutions: $(a,v)=(1,1),(0,0),(2,0)$. Are there any other rational solutions?
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Different solution of power Diophantine equation based on constant term
Let us define a power Diophantine equation by 2 algebraic functions $f,g$ (having different degree) and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $...
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On a variant of Brocard's problem using the definition of Pochhammer symbols
I've considered the following variant of Brocard's problem $$\frac{(2n-1)!}{(n-1)!}+1=m^2\tag{1}$$
for integers $n\geq 1$ and integers $m\geq 1$. I was inspired from the fact that the evaluation of ...
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Reference request: Markoff type equations
Consider the equation $x^2+ay^2+bz^2=(1+a+b)xyz.$ If there are infinitely many integral solutions, then up to permutations $(a,b)=(1,1),(1,2),(2,3).$ I have found a presentation by Waldschmidt (https:/...
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Solutions to linear equations from recurrence sequences with no repeated roots
Let $U=(u_n)_{n=0}^{\infty}\subseteq\mathbb{C}$ be a sequence enumerated by a linear homogeneous recurrence relation with constant coefficients, i.e., there is some $d\geq 1$ and $a_1,\ldots,a_d\in\...
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Upper bounds for solutions to a Pell-like equation
Let $N$ be a fixed positive integer that is not a square and $m$ be any nonzero integer. Let $x$ and $y$ be positive integers that solve $$x^2 - N y^2 = m^2$$ with $x + y$ minimal (in light of the ...
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Dependence on parameters of solvability of a non-linear Diophantine system
For which $(k,t)\in\mathbb Z^2\times\mathbb Z$ does there exist $(v,s)\in\mathbb Z^2\times\mathbb Z$ so that $|v|^2=s^2\neq0$ and $v\cdot k+st=0$?
I do not care what the solutions $(v,s)$ are but only ...
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How to solve the following system of diophantine equations? [closed]
We have the equations
$$a_1x+b_1y+c_1z=d_1,$$
$$a_2x^2+b_2y^2+c_2z^2=d_2,$$
$$a_3x^3+b_3y^3+c_3z^3=d_3,$$
where $a_i,b_i,c_i \in\Bbb N$ at $i \in \{1,2,3\}$ are known.
Is there an efficient ($O(\...
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A three variable linear diophantine promise problem
Given $a,b,c,s\in\Bbb N$ such that $(a,b,c)=1$ with promise that we have at most one triple $x,y,z\in\Bbb N$ such that $ax+by+cz=s$, what is a good algorithm that runs in $O(\log(abcs))$ time to find ...
user76479
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Erdős-Straus with 4 terms
The Erdős-Straus conjecture states that any fraction of the form $\frac{4}{n}$ can be decomposed as an Egyptian fraction with just 3 terms. In related research, I've recently come across conditions on ...
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How do you bound the exponent of x^2+1=y^p?
How do you bound the exponent of $x^2+1=y^p$ for $p$ a prime exponent using linear forms in logs?
So far I have $(x-i)(x+i)=y^p$ which are coprime and hence $x+i=(a+ib)^p$.
Now how do I get a linear ...
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A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
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Solution modulo $9$ of certain linear equation implies triviality modulo $3$
Question: Let $k \geq 2$ and $r \geq 4$ be two natural numbers. We are given eight integers $\nu_{ij} \geq 0$ for every $1 \leq i \leq k$ and $1 \leq j \leq r$ such that the following two conditions ...
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Conjecture about some recurrent primes
I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but ...
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Fermat degree FRM(n)
$\DeclareMathOperator\FRM{FRM}\DeclareMathOperator\frm{frm}$Assume $\ 3\le n\in\mathbb N$.
Question: What is the highest degree $\ f\in\mathbb N\ $ such that there does not exist any none-zero ...
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On the integer solutions of the equation $y^2 = x^3 + n$
Let $n$ be a nonzero integer. I am interested in the integer solutions $(x, y)$ to the equation $y^2 = x^3 + n$.
Let $S$ be the set of all integer solutions $(x, y)$ to this equation.
I am wondering ...
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Finding integral points of quadric without degree 1 terms
I consider for some $n\in\mathbb{N}$ the index set $I\subset\binom{n}{2}$ the following polynomial $p_I\in\mathcal{R}:=\mathbb{R}[x_1,...,x_n]$ with
$$p_I(x_1,...,x_n)=\sum_{\lbrace i,j\rbrace \in I}(...
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On the (hyper?)elliptic curve $y^2=x^2-x^3z^2+z-1$
The question here is if there exists $x,y,z\in\mathbb Z$ such that$$y^2=x^2-x^3z^2+z-1\label{1}\tag{1}$$other than the trivial solution$$x=0,y^2+1=z\text{ for all }y\in\mathbb Z\label2\tag2$$I know ...
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Elementary method for finding integer solutions for certain types of elliptic curve
There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $...
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Number of solution to homogeneous linear Diophantine equations
Let $T,M\in\mathbb{N}$ be fixed. Consider a linear Diophantine equation of the form
$a_1 x_1 + a_2 x_2 + … + a_n x_n = 0 $
with $a_i \in [-T,T] \subset \mathbb{Z}$. Is there an asymptotic formula to ...
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Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed
The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
- 3,098
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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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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 ...
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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 ...
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The strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation
Let us consider the strong twin conjecture:
For all positive integer $n$ there exist a prime $p$ such that
$$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime
Since the inequalities and the ...
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conjecture about quadratic equation
All the variables I deal with are integers. Given $q \geq 2, |k| \geq 2$, impose a constraints that we have an integer $p$ such that $q^2=1+pk=1\pmod k.$ For example, $3^2=1+1\cdot8, 5^2=1+3\cdot8.$ ...
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A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem
I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
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The number of solutions of $2^xpx+k=y^2$
Let's consider the family of diophantine equations
$$2^xpx+k=y^2$$
being $p\gt2$ a prime and $k$ a positive integer.
An example is given by the equation
$$2^x\cdot3x+97=y^2$$
that presents, at least, ...
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Is there a permutation $\tau\in S_n$ with $\tau(1)^{\tau(2)}+\cdots+\tau(n-1)^{\tau(n)}+\tau(n)^{\tau(1)}$ a square?
Let $n>1$ be an integer, and let $S_n$ be the symmetric group of all the permutatins of $\{1,\ldots,n\}$.
I'm curious whether there is a permutation $\tau\in S_n$ such that
$$\tau(1)^{\tau(2)}+\...
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A conjectural limit involving primorial and factorial
It is well known that the abc conjecture implies that the there are only finitely many solutions to Brocard problem, as shown by Overholt in Overholt, Marius (1993), "The diophantine equation $n! ...
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Genus $0$ algebraic curves integral points decidable?
It is known there is an explicit algebraic variety in $\mathbb Z[x_1,\dots,x_t]$ a bounded $t>2$ whose integral zero-set is non-empty is undecidable.
If the variety has genus $0$ is there anything ...
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A certain Pell Equation
Recently I came up with a positive solution $((x,y)\neq (\pm 1;0))$ to this diophantine equation
$$
x^2-\left(w^2(2^{n-2}p)^2+2^n(2^{n-2}p)\right)y^2=1,\qquad n\geq 2,
$$
where all variables are in $ ...
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Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$
I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$.
Lattice point means integer coordinates and equation with integer means diophantine ...
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Maximum number of integer solutions with some size constraints to bivariate polynomials?
Take a bivariate polynomial of total degree $d$ satisfying $d=d_x=d_y>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
Given a ...
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On the maximum number of common integer solutions in a system of polynomials
If we have $n$ degree $d$ algebraically independent polynomials in $n$ variables in $\mathbb Z[x_1,\dots,x_n]$ then
what is the maximum number of common integer solutions the system can have?
same ...
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Common integer roots of polynomials
I have two polynomials of form
$$f_1(w,x)=M_1$$
$$f_2(y,z)=M_2$$
and I have two polynomials of form
$$g_1(w,x,y,z)=M_3$$
$$g_2(w,x,y,z)=M_4$$
where $f_1,f_2,g_1,g_2\in\mathbb Z[w,x,y,z]$ and $M_1,M_2,...
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Can someone explain to a "newbie" of number theory how Matijasevič demonstrated the impossibility of hilbert’s tenth problem?
As the title stated, I'm an amateur in the number theory that has just approached hilbert's tenth problem and the demonstration given by Matijasevic, but I couldn't find much on it, and what I could ...
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On Sums of powers II
In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is ...
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Chinese remaindering to solve solvable diophantine equations
Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...
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When is $\phi(a^n+b^n+c^n)=0\mod n$?
A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
user147204
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Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples
$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.
Is it true that there are ...
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Number of integer solutions to quadratic polynomial with integer coefficients
It is known from for example Representations of Integers as Sums of Squares by Grosswald, E. that
$$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
- 193
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On variants of the abc conjecture in terms of Lehmer means
In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$
see the reference Wikipedia Lehmer mean.
The ...
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Quadratic factors of $l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$
Related to sum of three squares and this question.
Let $l_1,l_2,l_3 \in \mathbb{Z}[x,y]$ and $n \in \mathbb{Z}$.
Assume that $n$ is not a cube and not twice cube.
Let $f=l_1(x,y)^3+l_2(x,y)^3+l_3(x,...
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On the number of solutions of the equation involving Pochhammer symbols $(n)_a\cdot(n)_b=(n)_c$, for integers greater than or equal to $2$
As paticular case of the equation involving Pochhammer symbols $$(n)_a\cdot(m)_b=(k)_c,$$
where the variables are positive integers, I've consider the case $n=m=k$ of previous equation, that is
$$(n)...
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What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
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Find solutions and get a first statement for these diophantine equations inspired in certain figurate numbers
Yesterday I was thinking in speculative relationships between certain figurate numbers, please see if you need the tables, and references from the article of the
encyclopedia MathWorld Figurate Number....