# Welcome to MWiki

## Theorems of the month

### Cube Theorem

A sum $m \in {}^{\omega }{\mathbb{Z}}$ consists of three cubes for $a, b, c, n \in {}^{\omega }{\mathbb{Z}}$ if and only if

$m=n^3 + (n + a)^3 + (n - b)^3 = 3n^3 + a - b + 6c \ne \pm 4\mod 9$

is true. This implicitly quadratic equation yields the formula to be satisfied by $n.\square$

### Fickett's Theorem

For any relative positions of two overlapping congruent rectangular $n$-prisms $Q$ and $R$ with $n \in {}^{\omega }\mathbb{N}_{\ge 2}$, it can be stated for the exact standard measure $\mu$, where $\mu$ for $n = 2$ needs to be replaced by the Euclidean path length $L$, that:

$1/(2n - 1) < r := \mu(\partial Q \cap R)/\mu(\partial R \cap Q) < 2n - 1.$

#### Proof:

Since the underlying extremal problem has its maximum for rectangles with the side lengths $s$ and $s + 2d0$, min $r = s/(3s - 2d0) \le r \le$ max $r = (3s - 2d0)/s$ holds. The proof for $n > 2$ is analogous.$\square$