Multiplication Table (Quadratics Part I)

Abstract: This study develops a geometric framework for integer multiplication based on the Full Multiplication Table (FMT) and its minimal representative, the Triangular Multiplication Table (TMT). The FMT reveals a global polynomial architecture: every diagonal sequence is of the form , exhibits constant second differences, and belongs to a unified family of square-minus-square and oblong-minus-oblong curves. These curves appear as parabolic bundles that share a common axis and encode the quadratic symmetries of multiplication. Restricting the FMT to the first octant produces the TMT, a domain that contains all essential multiplicative geometry while eliminating redundancy. Within the TMT, the square and oblong families intersect each odd column in a highly structured way. When the vertical range is restricted to , each column is either “covered’’ by at least one square or oblong intersection (product) or “uncovered’’ by both. These two geometric outcomes correspond exactly to composite and prime values of , respectively. Thus, the classical arithmetic distinction between primes and composites emerges from a purely geometric covering mechanism driven by the quadratic structure of the multiplication table. The analysis establishes the parabolic loci of squares and oblongs, explains their shared symmetry axis , and formulates the covering criterion that underlies the square–oblong sieve.

Keywords: Multiplication Table; Triangular Multiplication Table (TMT); Full Multiplication Table (FMT); Quadratic Sequences; Square-minus-square numbers; Oblong-minus-oblong numbers.

2020 Mathematics Subject Classification: 11B85 – Arithmetic progressions; sequences in general ; 11B83 – Special sequences and polynomials (e.g., square numbers, oblong numbers); 11A25 – Arithmetic functions; related numbers; 97F60 – Mathematics education: Number theory.