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Theorems of the month

Cube Theorem

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

[math]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]n.\square[/math]

Fickett's Theorem

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


[math]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]s[/math] and [math]s + 2d0[/math], min [math]r = s/(3s - 2d0) \le r \le[/math] max [math]r = (3s - 2d0)/s[/math] holds. The proof for [math]n > 2[/math] is analogous.[math]\square[/math]

Recommended reading

Nonstandard Mathematics