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 ...
935 questions
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What is the smallest unsolved Diophantine equation?
If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
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Diophantine equation for generating computably enumerable set
By Matiyasevich's theorem, each member of computably enumerable set can be obtain from a diophantine equation system. For prime numbers, this system of diophantine equation is found. My question is:
...
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Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?
It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since
$$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,...
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On segments of the series $\sum_p\frac1{p-1}$
Here I ask a question concerning segments of the divergent series
$$\sum_p\frac1{p-1}=\sum_{k=1}^\infty\frac1{p_k-1},\tag{$*$}$$
where $p$ runs over all the primes, and $p_k$ denotes the $k$-th prime.
...
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Is it true that $\sum_{k=m}^n\frac{\sigma(k)}k\not\in\mathbb Z$ for all derangements $\sigma\in S_n$ and $1\le m\le n$?
Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$.
Recall that a permutation $\sigma\in S_n$ is called a derangemnt if $\sigma(k)\not=k$ for all $k=1,\ldots,n$.
Motivated ...
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Is there a permutation $\pi\in S_n$ with $\sum\limits_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0$ for each $n>7$?
Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$.
QUESTION: Is it true that for each $n=8,9,\ldots$ we have
$$\sum_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0\tag{$*$}$$
for ...
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Derangements and unit fractions
Motivated by a recent question of Zhi-Wei Sun and its nice answer by Zhao Shen, here are two related questions.
Let $S_n$ be the group of permutations on $\{1, 2, \ldots, n\}$.
a. For each $n \ge ...
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Can the partition function $p(n)$ take perfect power values?
Recall that the perfect powers are those integers $m^k$ with $k,m\in\{2,3,\ldots\}$. I don't consider $0$ or $1$ as a perfect power.
Y. Bugeaud, M. Mignotte and S. Siksek [Annals of Math., 2006] ...
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Permutations $\pi\in S_n$ with $\sum_{k=1}^n\frac1{k+\pi(k)}=1$
Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$.
Motivated by Question 315568 (http://mathoverflow.net/questions/315568), here I pose the following question.
QUESTION: ...
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0
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Showing a rational polynomial is non injective
Let $p$ be a prime number greater than $11$, further let $x<y<z<2p$ be positive integers coprime to $p$, such that $x+y+z=3p$.
Is it possible to show that $f(x,y,z)=\dfrac{xy+xz+yz-2p^2}{...
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Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution?
Does there exist a positive integral solution $(x, y, n)$ to $(xy+1)(xy+x+2)=n^2$? If there doesn't, how does one prove that?
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A question regarding Goormaghtigh conjecture
I have a question regarding Goormaghtigh conjecture on the Diophantine equation
$$\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}.$$
Suppose that a positive integer $N$ is given. How many integer solutions are ...
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2
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On the existence of integer square root of a $3 \times 3$ positive definite matrix
As far as I know, a real square matrix $M$ has a real square root if $M$ is positive semidefinite, i.e., if all eigenvalues are nonnegative. And, in fact, its square root is unique.
I have read some ...
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Linear diophantine quasivariety having a unique solution
Consider the equation
$$6x+3y+2z=13$$
for $x$, $y$, $z$ nonnegative integers,
with the constraints
$$x=0\implies y=0,$$
$$x=0\implies z=0.$$
The set of solutions $(x,y,z)$ is a kind of quasivariety
...
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Does $2^x-3p^y=5$ (with $p$ an odd prime) have only finitely many positive integer solutions?
Let $p$ be an odd prime. Does the equation
$$2^x-3p^y=5$$
only have finitely many solutions in positive integers $x$ and $y$?
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Number of integer solutions of a linear equation under constraints
How many positive integer solutions of $$\sum_{i=1}^{k}x_i = N$$ for some positive integer $N$ given the constraints $n_i\leq x_i\leq m_i$ for $i=1,\ldots,k$, where $n_i$ and $m_i$ are positive ...
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Rational perfect power values of $y(y+1)$
This is hard, so I am looking for partial results and how hard it is.
Let $n>4$. Is it true that the hyperelliptic curve $x^n=y(y+1)$
doesn't have rational point with $x \ne 0$?
If necessarily ...
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2
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Hyperelliptic curves imply FLT-like results
Probably this is known, but doesn't show in searches.
If a certain hyperelliptic curve has only trivial rational points,
FLT-like curve also has only trivial rationals points for fixed $n$.
Working ...
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0
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No rational points on $x^n+a=y^2$ for all $n>4$"?
Is there rational (or better integer) $a$ such that for all $n>4$,$x^n+a=y^2$
has no rational points?
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1
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0
answers
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On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$
For $n>4$ and coprime integers $x,y,z$ consider the diophantine equations:
$$x^4+16z^n=y^2 \qquad (1)$$
and
$$x^4+z^n=y^2 \qquad (2)$$.
(2) is special case of Fermat Catalan and is solved.
For ...
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4
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2
answers
363
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A specific Diophantine equation restricted to prime values of variables.
Consider the following Diophantine equation $$x^2+x+1=(a^2+a+1)(b^2+b+1)(c^2+c+1).$$ Assume also that $x,a,b,c,a^2+a+1,b^2+b+1, c^2+c+1$ are all primes. We'll call such a quadruplet $(x,a,b,c)$ a ...
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Solving elliptic equation in rational functions
Good afternoon,
I'm trying to solve an elliptic equation of the form
$$AY^2=4X^3+aX+b$$
where $A\in\mathbb{C}[z]$, $a,b\in\mathbb{C}$ and the unknowns $X,Y\in\mathbb{C}(z)$. In Mason ``Diophantine ...
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What is the time complexity for solving Diophantine equations of degree 2?
Manders and Adleman mention that the computational complexity for binary quadratic Diophantine equations is NP-complete. Has a more specific complexity been claimed for polynomials of the form $Axy + ...
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A generalization of Bernoulli's inequality and what does it application for?
Let $a_1 \ge a_2 \ge \cdots \ge a_n \ge 1$ or $0 \le a_1 \le a_2 \le \cdots \le a_n \le 1$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 1$ then
$$\left(\sum_{i=1}^{n}{\alpha_i} \right)\...
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On the diophantine equations $x^n+n=y^m$ and $x^n-n=y^m$
Here I ask a question concerning the diophantine equations
$$x^n+n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\},\tag{1}$$
and
$$x^n-n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\}.\tag{2}$$
...
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$(2x^2+1)(2y^2+1)=4z^2+1$ has no positive integer solutions?
Equation
$$(2x^2+1)(2y^2+1)=4z^2+1$$
has no solutions in the positive integers. Its true?
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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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Isomorphism classes of lattices
Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define
$$
V = \{x \in \mathbb R^6 \mid A \cdot x = 0\}
$$
and
$$
\Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...
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votes
4
answers
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Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
For a fixed positive integer $n$, the Diophantine equation
$$x^2 + y^2 + z^2 = n$$
was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...
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2
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On the Diophantine equation $x^{4}+y^{4}=z^p$
Do there exist integers $x,y,z$ with $xyz\neq 0$, such that
$$x^4 + y^4 = z^p$$
where $p\geq 5$ is some prime ?
If yes, are there infinitely many of them ? And if there exists infinitely many of ...
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Does Fermat's last theorem hold in the Grothendieck ring of the ordinals?
Inspired by this question and its answer, I am curious whether or not Fermat's last theorem holds in the Grothendieck ring of the ordinals under Hessenberg (commutative) operations.
The excellent ...
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Enquiry on a Diophantine problem
Let $x,y, z$ be relatively prime integers with $xyz \neq 0$. Suppose that
$$x^{m/n} + y^{m/n} = z^{m/n}$$
where $m,n$ are relatively prime integers with $mn \neq 0$.
Does it necessarily follow ...
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Density version of the Erdős-Graham conjecture
In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdős-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into finitely many subsets then one of the subsets ...
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Number of nontrivial integral solutions to $f(x)=f(y)$
Let $f(x)\in\mathbb Z[x]$ be a nonconstant polynomial, and let $$g(x,y)=\frac{f(x)-f(y)}{x-y}\in\mathbb Z[x,y].$$ Let $N(B)$ denote the number of pairs of integers $(x_0,y_0)$ such that $1\le x_0,y_0\...
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Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$
Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...
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Dimension of $S$-units over $\mathbb{C}[x]$
Let $S=\{s_1,\ldots,s_n\}$ be a finite set of complex numbers. Consider the set of polynomials $$U=\{\,(x+s_1)^{k_1}\cdots(x+s_n)^{k_n}:\, 0\leq k_1+\cdots+k_n\leq H\}.$$
I am curious as to what is ...
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0
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Does each integer have the form $x^4-y^3+z^2$ with $x,y,z$ positive integers?
Let $\mathbb Z^+$ denote the set of positive integers. Here I ask the following question.
QUESTION: Does each integer have the form $x^4-y^3+z^2$ with $x,y,z\in\mathbb Z^+$?
I guess that the answer ...
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Solution to an exponential Diophantine equation
I am trying to solve the following exponential Diophantine equation:
$$ 9^{k_1} -2^{j_1} = 9^{k_2}-2^{j_2}$$
My conjecture is that this implies $k_1=k_2$ and $j_1=j_2$, apart from eventually some ...
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Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?
Recall that the triangular numbers are those integers
$$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$
In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...
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0
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Does every integer $n>1$ have the form $a^2+b^2+3^c+5^d$ with $a,b,c,d$ nonnegative integers?
Lagrange's four-square theorem states that every nonnegative integer is the sum of four squares. I have tried to replace two of the four squares by two powers. This leads to my following question: ...
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0
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300
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Mathematical Aspects of Hectoc-type Puzzles
hectoc is a puzzle, where one is given a sequence of six decimal digits and the task is to intersperse arithmetic operations from the given set $+,-,/,*$ and matching brackets $(,)$ in a way that the ...
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1
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Solutions to diophantine equation
I have been working on solutions to $x^5+y^5+z^5=1$, and I found that the three solutions of $x^3+bx+\frac{1}{5b}$ satisfy that equation. Multiplying by $5b$: $5xb^2+5x^3b+1=0$, then solving for b ...
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1
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168
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Diophantine equations and 'quasi-paucity'
Let $X,Y \geq 1$. I am interested in the number of solutions of the following diophantine equations:
$$S_1\colon \, \, x_1y_1^3 = x_2 y_2^3 $$
Let $N_1(X,Y) $ denote the number of solutions to $S_1$ ...
- 163
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0
answers
197
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Full-rank factorization property of integer-valued matrices
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\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\...
- 128k
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0
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Is every integer a difference of two powers?
True or false? (I don't know.) Every positive integer is the difference of two powers. Examples:
$ 1 = 3^2 - 2^3 $
$ 2 = 3^3 - 5^2 $
$ 3 = 2^7-5^3 $
$ 4 = 2^3-2^2 = 5^3-11^2 $
$ 5 = 2^5 - 3^3 $
...
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1
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760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
3
votes
1
answer
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What is known about equation $a^{n+k}+b^{n+l}=c^{n+m}$ and its set of solutions?
Suppose that $(k,l,m) \in \mathbb{N_0}^3$.
If $(k,l,m)=(0,0,0)$ then for $n=1,2$ there is an infinite number of solutions and, by the theorem of of Wiles there are no solutions when $n \geq 3$.
Is ...
- 513
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1
answer
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Solve this Diophantine equation $(2^x-1)(3^y-1)=2z^2$
Find the positive integers $(2^x-1)(3^y-1)=2z^2$ have three solutions
$$(1,1,1),(1,2,2),(1,5,11)$$I already know $(2^x-1)(3^y-1)=z^2$ has no solution. See: P.G.Walsh December 2006 [On Diophantine ...
- 4,280
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2
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1k
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How are such sets of natural numbers called?
I heard about this problem an year ago, but I just can't remember the name.
The problem goes like this: study the sets
$\{a_1,a_2,\dotsc,a_m\}\subseteq\mathbb{N}$ such that if $1\leq i<j\leq m$,...
- 103
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0
answers
223
views
Product of four consecutive primes plus $1$ equals square
Some days ago, I noticed that $3\cdot 5\cdot 7\cdot 11 +1=34^2$.
I am almost sure that if we denote four consecutive primes by $p, q, r, s$ then the equation
$$p\cdot q\cdot r\cdot s+1=x^2 \quad ...
- 3,866