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

Abstract

Whether a 3 ×3 magic square can be built from nine distinct perfect squares is a celebrated open problem of LaBar and Gardner. Relaxing it by one line yields Lee Sallows’ classical near- miss: nine distinct squares whose three rows, three columns and one diagonal share the common sum 21609 = 1472 = 9·74. We call such an array a seven-line square. We give an explicit two- parameter family of seven-line squares whose common sum is 9 m4 with m= a2 +ab+b2, together with a complete symbolic verification of all its defining identities. The family unifies the two previously published one-parameter constructions: Ant King’s (2011) is the slice b= a−1 and Benjamin Bartsch’s (2017) is the slice b= 1. Because a2 + ab+ b2 is the Eisenstein (L¨oschian) norm, the family realizes the constant 9p4 for every prime p≡1 (mod 3) — including infinitely many primes (e.g. 67,79,97,103,109) lying in neither the King nor the Bartsch slice. This proves the existence half of the equivalence “9p4 is a seven-line constant iff p≡1 (mod 3)”; we also prove, for the whole family, that it never completes to a full magic square of squares. The remaining half — necessity, that p≡2 (mod 3) admits no such square — is shown to be free of any congruence obstruction, hence genuinely arithmetic (of congruent-number / elliptic-curve type); we leave it as a conjecture, verified for every prime p≤113.