Main Page
Welcome to MWiki
Theorems of the month
Three-Cube Theorem
By Fermat’s little theorem, [math]\displaystyle{ k \in {}^{\omega }{\mathbb{Z}} }[/math] is sum of three cubes if and only if
and [math]\displaystyle{ a, b, c, d, m, n \in {}^{\omega }{\mathbb{Z}} }[/math] implies both [math]\displaystyle{ (a^2 + b^2)n - (a - b)n^2 = c =: dn }[/math] and [math]\displaystyle{ m^2 = n^2 - 4(b^2 - bn + d) }[/math] for [math]\displaystyle{ 2a_{1,2} = n \pm m.\square }[/math]
Fickett's Theorem
For any relative positions of two overlapping congruent rectangular [math]\displaystyle{ n }[/math]-prisms [math]\displaystyle{ Q }[/math] and [math]\displaystyle{ R }[/math] with [math]\displaystyle{ n \in {}^{\omega }\mathbb{N}_{\ge 2} }[/math] and [math]\displaystyle{ m := 2n - 1 }[/math], it can be stated for the exact standard measure [math]\displaystyle{ \mu }[/math], where [math]\displaystyle{ \mu }[/math] for [math]\displaystyle{ n = 2 }[/math] needs to be replaced by the Euclidean path length [math]\displaystyle{ L }[/math], that:
Proof:
Since the underlying extremal problem has its maximum for rectangles with the side lengths [math]\displaystyle{ s }[/math] and [math]\displaystyle{ s + 2d0 }[/math], min [math]\displaystyle{ r = s/(3s - 2d0) \le r \le }[/math] max [math]\displaystyle{ r = (3s - 2d0)/s }[/math] holds. The proof for [math]\displaystyle{ n > 2 }[/math] is analogous.[math]\displaystyle{ \square }[/math]