An Infinite Family of Near-Magic Squares of Squares: the 9p4 Conjecture

Abstract

The existence of a 3 ×3 magic square whose nine entries are distinct perfect squares is a celebrated open problem. Relaxing the requirement by one line yields Lee Sallows’ classical example, a 3 ×3 array of nine distinct squares whose three rows, three columns, and one diagonal share the common sum 21609 = 1472. We call such an array a seven-line square and its common sum a seven-line constant. Using two purpose-built programs—a fertility finder and a completability verifier employing an optimized constraint-propagation mechanism—we discover that among integers of the form N = 9p4 with p prime, a seven-line square exists if and only if p ≡1 (mod 3), and that no member of this family ever completes to a full magic square of squares (the eighth line always fails). Sallows’ example is the first member (p = 7). We record these observations as two conjectures, verified by exhaustive computation for every prime p ≤113, and we relate the congruence p ≡1 (mod 3) to the splitting of p in the ring of Eisenstein integers.