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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 ...

10
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1answer
194 views

Universality of $y^4-x^3$ mod $p$

For pedagogical reasons, I got interested in the equation $y^4-x^3=a$ over $\mathbf F_p$. To my surprise (maybe I'm naive), there is only one couple $(p,a)=(13,7)$ for which there is no solution, at ...
2
votes
3answers
415 views

Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)

Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?
5
votes
1answer
164 views

Counting primitive solutions to a diophantine inequality

This is a refinement (perhaps a simpler version) of a question I asked here before and couldn't get an answer for. Fix $\alpha \in (0,1]$ and a small constant $c>0$. For $x \in [0,1]$ and $N\in\...
2
votes
0answers
140 views

Is there a general method for solving Diophantine equations of this type?

Is there a general method for solving Diophantine equations of the form $${x_1}^n + {x_2}^n+ \cdots + {x_m}^n ={x_{m+1}}^n$$ where $x_{i}\geq 1, m\geq 3$ and $n\geq 2$ are integers ? I would also be ...
1
vote
1answer
220 views

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 $$x^3+y^3=3z^3$$
2
votes
0answers
130 views

trivial solutions for Diophantine equations

Let $K$ be an odd degree number field. Consider the Diophantine equation: $$ X^4 + bY^4 =Z^2 $$ where $b\neq 0$. Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
15
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0answers
593 views

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 ...
2
votes
1answer
90 views

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: ...
21
votes
0answers
291 views

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,...
0
votes
0answers
128 views

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. ...
0
votes
1answer
151 views

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 ...
4
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0answers
125 views

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 ...
1
vote
1answer
154 views

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 ...
6
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0answers
116 views

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] ...
19
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1answer
612 views

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: Is ...
2
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0answers
189 views

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}{...
17
votes
2answers
719 views

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?
2
votes
1answer
298 views

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 ...
3
votes
1answer
198 views

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 ...
5
votes
0answers
82 views

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 ...
9
votes
1answer
271 views

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 $y$?
5
votes
1answer
530 views

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 ...
7
votes
1answer
205 views

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 ...
1
vote
2answers
269 views

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 ...
5
votes
0answers
169 views

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?
1
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0answers
115 views

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 ...
3
votes
1answer
250 views

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 ...
3
votes
1answer
97 views

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 ...
5
votes
1answer
185 views

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 + ...
2
votes
0answers
125 views

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)\...
1
vote
0answers
161 views

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 $$x^n-n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\}.\tag{2}$$ ...
7
votes
1answer
549 views

$(2x^2+1)(2y^2+1)=4z^2+1$ has no positive integer solutions?

Equation $$(2x^2+1)(2y^2+1)=4z^2+1$$ has no solutions in the positive integers. Its true?
0
votes
1answer
99 views

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\...
5
votes
0answers
181 views

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 ...
6
votes
4answers
417 views

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 ...
5
votes
1answer
214 views

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 ...
5
votes
1answer
269 views

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 ...
9
votes
1answer
468 views

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 ...
3
votes
1answer
176 views

Density version of the Erdos-Graham conjecture

In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdos-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into finitely many subsets then one of the subsets ...
4
votes
0answers
131 views

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\...
0
votes
0answers
49 views

Solutions to exponential diophantine: 2^a + 3^b = 2^c + 3^d [duplicate]

I am interested in the finding all the solutions to the equation in the subject: 2^a + 3^b = 2^c + 3^d All I have found online is the Pacific Journal of Mathematics 1982 Vol. 101, No. 2 "Exponential ...
2
votes
0answers
193 views

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 ...
1
vote
1answer
115 views

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 ...
2
votes
0answers
182 views

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 ...
3
votes
1answer
155 views

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 ...
5
votes
0answers
132 views

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, ...
15
votes
0answers
428 views

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: ...
0
votes
0answers
67 views

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 ...
4
votes
1answer
198 views

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 ...
3
votes
1answer
139 views

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 $S_1$ ...