AdditionTable

The addition table of ℤ/n, R = QuotientRing(Integers, n): the matrix whose (i, j) entry is aᵢ + aⱼ, its rows and columns ℤ/n's elements in the order ElementOrder lists them.

AdditionTable(R)the n × n matrix of sums, as ResidueClass entries, for 2 ≤ n ≤ 64

AdditionTable(R, ElementOrder -> order)rows and columns listed Natural (0, 1, …), by their ChineseRemainder residues, or by their p-adic digits (Adic)

Typed signatures

Type	Package	Applies
(value, value*) -> list	residues	always

Domain: Modular arithmetic

Details

• A circulant: each row is the one above shifted by one place, so every value appears once in each row and column
• Listed by Chinese remainders, ℤ/mn's table (m, n coprime) is ℤ/m's with a copy of ℤ/n's in each cell; listed by digits, ℤ/pᵏ's is p × p blocks of ℤ/pᵏ⁻¹'s
• A Show draws it at any size as ArrayPlot(AdditionTable(R)), its cells colored by their values; as a value it stops at n = 64

Examples

1 more kept as data, checked against the oracles

Basic #

ℤ/3: each row the one above, shifted

live ↗

In$\mathrm{AdditionTable}(ℤ/3ℤ)$Out$\begin{pmatrix}\overline{0}_{3} & \overline{1}_{3} & \overline{2}_{3}\\ \overline{1}_{3} & \overline{2}_{3} & \overline{0}_{3}\\ \overline{2}_{3} & \overline{0}_{3} & \overline{1}_{3}\end{pmatrix}$

listed by binary digits, 0, 2, 1, 3: two blocks by parity, each a copy of ℤ/2's table

live ↗

In$\mathrm{AdditionTable}(ℤ/4ℤ, \mathrm{Rule}(\mathrm{ElementOrder}, \mathrm{Adic}))$Out$\begin{pmatrix}\overline{0}_{4} & \overline{2}_{4} & \overline{1}_{4} & \overline{3}_{4}\\ \overline{2}_{4} & \overline{0}_{4} & \overline{3}_{4} & \overline{1}_{4}\\ \overline{1}_{4} & \overline{3}_{4} & \overline{2}_{4} & \overline{0}_{4}\\ \overline{3}_{4} & \overline{1}_{4} & \overline{0}_{4} & \overline{2}_{4}\end{pmatrix}$

About

Examples	3

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram's GroupMultiplicationTable[CyclicGroup[n]] tabulates the same group, by element indices in GroupElements' order rather than by residues.

See also: MultiplicationTable, QuotientRing, ResidueClass