## New answers tagged diophantine-equations

0
votes

### Almost Pell type equation

If you multiply this by $2$ you get $(2x)^2 - 2N y^2 = -2$, so you can solve $x^2 - 2N y^2 = -2$ and then check when $x$ is even. $x^2 - 2Ny^2 = -2$ is a generalized Pell's equation, so you can find a ...

6
votes

Accepted

### Representation of a number as a product of $\sqrt{n^2 + 1} + n$

$\def\supp{\mathop{\mathrm{supp}}}$ Surely, in the ``Almost equivalent question'' you assume that the $d$'s are square-free.
Denote $R=\mathbb Z[\sqrt{p_1},\dots,\sqrt{p_n}]$. We prove that in its ...

4
votes

### Representing $x^6-4$ as a sum of two squares

This problem looks tricky. I'd recommend you look up "Châtelet surfaces"; these are a slightly easier case when one has a polynomial of degree $4$ instead of a polynomial of degree $6$, but ...

3
votes

### Representing $x^3-2$ as a sum of two squares

One more way to solve the problem. Let $x = 4t + 3$. Then
$$x^3 - 2 = 16t^2(4t + 9) + (108t + 25).$$
The system
$$4t + 9 = a^2 \qquad 108t + 25 = b^2$$
has infinitely many solutions. It is reduced to ...

0
votes

### What is the taxicab number for rational fourth powers?

One appeoach is to seek numbers $N$ having the form
$N=a^4+b^4=\dfrac{c^4+d^4}{e^4}$
where all variables are $\in\mathbb{Z}_{>0}$ and $e>1$. Since $e$ must have only prime factors of the form $...

12
votes

### What is the taxicab number for rational fourth powers?

It has been known since Euler that the quartic surface defined by
$\displaystyle x_1^4 + x_2^4 = x_3^4 + x_4^4$
contains a rational curve defined by
$\displaystyle x_1(t) = t^7 + t^5 - 2t^3 + 3t^2 + t,...

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