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Theorems of the month

Cube Theorem

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

[math]\displaystyle{ m=n^3 + (n + a)^3 + (n - b)^3 = 3n^3 + a - b + 6c \ne \pm 4\mod 9 }[/math]

is true. This implicitly quadratic equation yields the formula to be satisfied by [math]\displaystyle{ n.\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], 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:

[math]\displaystyle{ 1/(2n - 1) < r := \mu(\partial Q \cap R)/\mu(\partial R \cap Q) < 2n - 1. }[/math]

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]

Recommended reading

Nonstandard Mathematics