Multiplication Table Modulo N (MTMN)

Definition

For a fixed integer $N \geq 2$, the multiplication table modulo $N$ is the array whose $(x,y)$-entry is $$ xy \bmod N, \qquad 1 \leq x,y \leq N-1. $$ We abbreviate this object by MTMN.

Each residue label $a \in \{0,1,\ldots,N-1\}$ determines the lattice set A_N_a, and the main geometric quantities are $$ S(N,a) := \operatorname{Area}(\operatorname{conv}(A_{N,a})), \qquad S(N) := \sum_{a=0}^{N-1} S(N,a). $$ So MTMN starts from finite arithmetic data and develops it into residue classes, convex polygons, and area invariants.

Core Reading Path

The main public route through the subject is:

  1. the residue classes A_N_a
  2. the per-residue and total areas S_N_a and S_N
  3. the solved composite-side zero class zero_class_geometry
  4. the exact boundary models first_boundary_model and second_boundary_model

Two early structural contrasts

Two contrasts organize the current theory from the start.

First, prime and composite moduli behave differently. Prime nonzero residue classes form the clean permutation-plot case, while composite moduli split into a unit-side geometry and a visible zero-divisor geometry. The zero class is the first place where the composite side becomes completely explicit; see zero_class_geometry.

Second, the table is naturally built border by border. Those visual layers already produce the exact lower models first_boundary_model and second_boundary_model.

Figures

Multiplication table modulo $5$ with convex hulls of all residue classes. This is the basic picture from which the subject starts.

Prime and composite moduli already separate into different geometric regimes.

The first two boundary layers already form an exact geometric pipeline rather than an ad hoc truncation.

Across moduli, the overlay of residue-class hulls suggests both global growth and arithmetic sensitivity.

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