Theory
Theory contents
The published chapters proceed from the general linear structure of a magic square to arithmetic and Diophantine restrictions. Each chapter separates proved statements from open questions.
01→02→03→04→
The general form of a 3×3 magic square
Proofs of M=3E and the existence and uniqueness of m(E,x,y), with exact versions over groups, rings, fields, ℤ, ℚ, and ℝ.
Residues and quadratic residues
Coordinate congruence, restrictions modulo 3, 4, 5, and 24, and an application of the Legendre symbol to prime divisors of entries.
The magic square of squares problem
The 9/9 and 7/9 problems, the known example, and the transition from the linear normal form to Diophantine constraints.
General theory of the 4/9 and 5/9 patterns
D₄ orbits, elimination of E,x,y, quadratic systems, colored supports, and criteria for completeness of parametrizations.