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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**4**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**6**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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}) \...