Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 $\mathbb{Q}$. ...

2
votes
0answers
130 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
690 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?
0
votes
0answers
132 views

On the finite field analogue of the Erdos-Moser conjecture

The Erdos-Moser conjecture is the statement that, if $m,k$ are positive integers and $$1^{k} + 2^{k} + \cdots + m^k =(m+1)^k,$$ then $(k,m)=(1,2)$. Is there a finite field analogue of this ...
2
votes
1answer
270 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 ...
2
votes
1answer
174 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 ...
4
votes
0answers
79 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
270 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
470 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
202 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
265 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
165 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
vote
0answers
112 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
248 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
95 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
173 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
116 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
159 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
517 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
98 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
179 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
410 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
205 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
264 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
164 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
130 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
48 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
186 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
175 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
154 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
131 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
411 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
61 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
195 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 $S_1$ ...
2
votes
0answers
75 views

Full-rank factorization property of integer-valued matrices

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\...
10
votes
0answers
342 views

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 $ ...
11
votes
1answer
372 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 ...
0
votes
0answers
84 views

On existence of certain primes and integers

Given $N\gg 0$ and small $\epsilon>0$ take coprime $A,B\approx N^{\frac14+\epsilon}$. From exponential sums we can show that for every prime $p\approx N$ (but $>N$) there is an $m\in\Bbb Z$ such ...
3
votes
1answer
142 views

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 ...
6
votes
0answers
449 views

Solve this Diophantine equations $(2^x-1)(3^y-1)=2z^2$

Find the positive integers $(2^x-1)(3^y-1)=2z^2$ have three solution $$(1,1,1),(1,2,2),(1,5,11)$$I already know the solution of $(2^x-1)(3^y-1)=z^2$ has no solution.see:P.G.Walsh December 2006 On ...
8
votes
2answers
768 views

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 $1\leq i<j\leq m$,...
8
votes
0answers
141 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 ...
2
votes
0answers
111 views

Genus Zero Diophantine Equations and Infinite Valuations

I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem: [1] Solving genus zero diophantine ...
0
votes
6answers
541 views

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 ...
0
votes
0answers
146 views

Could a full rank linear system arise from this construction?

Fix a small ${\epsilon}>0$ and take a very large $n>0$. We say integers $A,B,C,D,A',B',C',D'$ satisfy property $P[n]$ if $A,B$ is coprime, $C,D$ is coprime, $A',B'$ is coprime and $C',D'$ is ...
2
votes
0answers
126 views

minimum size of undecidable quadratic diophantine problems

According to Matiyasevich, the existence of integer solutions of systems of polynomial equations with integer coefficients is undecidable. By introducing additional variables substituting factors of ...
16
votes
3answers
413 views

Number of solutions to polynomial congruences

Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that ...
2
votes
0answers
93 views

Bound for the number of solutions to a system of congruence relations

Suppose I have $n$ polynomials in $n$ variables $G_j(x_1, \ldots, x_n)$ with integer coefficients. Let $u_j$ be some fixed $p$-adic integers. Consider the system of congruences $$ G_j(\mathbf{x}) \...