# Tagged Questions

**1**

vote

**0**answers

136 views

### Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions ...

**1**

vote

**1**answer

201 views

### On $x^3-y^2=1728 \text{ unit}$ in number fields

Consider solution of
$$x^3-y^2=1728 \text{ unit} \qquad (1)$$
in a number field.
This is related to the discriminant of elliptic curve
in terms of $c_4,c_6$.
Via elliptic curves it might have ...

**20**

votes

**3**answers

628 views

### Consecutive square values of cubic polynomials

Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square?
It is known that the ...

**12**

votes

**1**answer

430 views

### State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
...

**0**

votes

**1**answer

428 views

### Like Diophantine equation

Dear all,
I have posted this question on m.s.e. Unfortunately, no one responded to answer. I hope, this site and members of this site will answer my questions.
The equation $x^n - ny^x-nxy$ = $0$ ...

**4**

votes

**2**answers

375 views

### Are there Heronian triangles that can be decomposed into three smaller ones?

Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...

**8**

votes

**1**answer

496 views

### how many consecutive integers $x$ can make $ax^2+bx+c$ square ?

The following problem was raised in a Mathlinks thread:
If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ?
The ...

**5**

votes

**5**answers

915 views

### Impossible Heronian Triangles (Ratio of 2 Sides)

There is no Heronian triangle (or simply consider triangles on an integer lattice
which also have integer side lengths) for which one side is half the length of
another side. What other "side-side ...

**2**

votes

**0**answers

484 views

### Does the following Diophantine equation have nontrivial rational solutions?

Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all?
If I am ...

**11**

votes

**5**answers

1k views

### Analysis of a quadratic diophantine equation

Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, ...

**10**

votes

**1**answer

773 views

### Fermat's Bachet-Mordell Equation

Fermat once claimed that the only integral solutions to $y^2 = x^3 - 2$ are $(3, \pm 5)$.
Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call ...

**11**

votes

**2**answers

1k views

### Transforming a Diophantine equation to an elliptic curve

I heard that the following problem lead to determine the rational points of an elliptic curve:
For which integers $n$ there are integers $x,y,z$ such that $x/y+y/z+z/x=n$. Could anyone show me why ...