All Questions

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers. Q1. Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...

Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$) of radius $r < 1$ avoid all rational points of height $\le h$? A rational point is a point all of whose coordinates are ...

Let $S(r)$ be the surface of the origin-centered sphere in $\mathbb{R}^3$. A point is an integer point if all its coordinates are integers. What is the smallest radius $r_n$ such that $S(r_n)$ ...

Is it the case that every unit-radius circular spiral, $$x = \cos(t)$$ $$y = \sin(t)$$ $$z = c \cdot t$$ for $c \in \mathbb{R}^+$ is dense in rational-coordinate points (i.e., all three coordinates ...

According to several sources, it is conjectured (or at least believed) that the rational points of curves over the rationals of genus $g > 1$ are uniformly bounded by $g$. E.g. here p. 1. Assuming ...

Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$. Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$. $C$ is birationally equivalent to $f(x)=y^2$. The constant coefficient of $g(x)$ is $1$ since $f$ is monic and $(...