The problem is to find the numbers $G_i$ such that the area inside the curve $|x|^{G_i}+|y|^{G_i}=r^{G_i}$ is an integer multiple of $r^2$. Because the curve defined by this equation is always symmetric by the x and y axis, it is equivalent to finding $G$ such that $F(r,G)=4 \int_0^r (r^{G}-x^{G})^\frac{1}{G} dx = i\ r^2, i \in \mathbb{N}$.
Here are some properties of this function:
$F(r,2)=\pi r^2$, since it is the area of the curve $x^2+y^2=r^2$, which is a circle.
$G_2=1$, since the curve $|x|+|y|=r$ defines a square which has side $\sqrt{2} r$, so its area is $2r^2=F(r,G_2)$.
$G_4=\infty$, since $\lim_{G\to\infty} F(G)=4r^2$. This limit is easy to prove by showing the curve $|x|^{G}+|y|^{G}=r^{G}$ converges pointwise into a square with side $2r$ when $G\to\infty$, so its area approaches $4 r^2$.
$G_0 = 0$, since $\lim_{G\to0} F(G)=0$. Again, the proof of this is made by showing that the curve $|x|^{G}+|y|^{G}=r^{G}$ converges pointwise into $y=0$, except for $x=0$.
The numbers $G_1$ and $G_3$ exist because the function $F(r,G)$ is continuous and because of the two limits above.
The numbers $G_j$ when $j>4$ do not exist.
Here is the approximated values I got for $G_1$ and $G_3$ using Newton's Method.
$$G_1=0.6072483224858489172223019262209528333...$$ $$G_3=1.7914738498700301966598792231223519090...$$
I used Wolfram Mathematica to try and calculate a closed form expression for $F(r,G)$. Here is what I got, without being able to simplify any further.
$$\frac{F(r,G)}{4 r^2} = \frac{\Gamma(1+\frac{1}{G})^2}{\Gamma(1+\frac{2}{G})} = \frac{(\frac{1}{G}!)^2}{\frac{2}{G}!}$$
$$\frac{F(r,G)}{4 r^2} = \frac{B(\frac{1}{G},1+\frac{1}{G})}{G}$$
$$\frac{F(r,G)}{4 r^2} = H(-\frac{1}{G},\frac{1}{G},1+\frac{1}{G},1)$$
$\Gamma$ is the gamma function
$B$ is the beta function (Euler integral of the first kind)
$H$ is the hypergeometric function
Questions directly related to the subject:
- Is there a name for a curve of the form $x^G+y^G=r^G$?
- Is there any meaningful relation between $G_1$ and $G_3$?
- Is there an easy way to check if either $G_1$ or $G_3$ is related to other famous numbers $\pi$, $e$, $\gamma$, $\phi$, ... I already checked OEIS for some common expressions, such as $G_1$, $G_1^2$, $\frac{1}{G_1}$, $\ln(G_1)$, $e^{G_1}$, $1 - G_1$, ...
- Is there any way to isolate $G$, in other words, find the inverse function $G = F^{-1}(F(r,G))$.
- Is there a way to check if either $G_1$ or $G_3$ is irrational or transcendental.
Questions about Mathematics in general, but related to the subject:
A. What area(s) of Mathematics is responsible for studying these type of things?
B. Any book recommendations that might help me understand the meaning of the functions H, B and $\Gamma$?
Obs: I also tried to find patterns in the partial fraction expansion of $G_1$ and $G_3$ with no luck. Also tried to get the rational approximations for each of them using the partial fractions and could not see any patterns or get any good rational approximations such as $\frac{22}{7}$ for $\pi$.