Consecutive Strict-Emirp Prime Magic Squares: Computational Constructions of Orders 9 to 19,683

Abstract

This preprint presents a computational research programme investigating magic squares constructed from consecutive members of the increasing sequence of strict emirp primes. A strict emirp is a prime number whose decimal digit reversal is also prime and differs from the original number. The study documents reported constructions at eight orders: 9, 27, 81, 243, 729, 2187, 6561 and 19,683. The largest reported construction comprises 387,420,489 entries. Unlike digit-reversible magic squares, these constructions do not require the reversed entries to form another magic square. Instead, the defining challenge is to arrange a consecutive sequence of distinct strict emirp primes so that every row, column and both principal diagonals share an identical magic constant. The paper establishes mathematical definitions, necessary arithmetic divisibility conditions and exact computational verification criteria. It also examines the relationship between prime-sequence selection, combinatorial arrangement and the increasing computational demands of larger magic squares. Seven spreadsheet or audit packages support the reported constructions through order 6561. The order-19,683 solution is additionally reported, with its complete numerical dataset and verification certificate awaiting incorporation. Independent verification of all eight constructions has not yet been completed for this manuscript. The work represents an ongoing investigation into computational number theory and combinatorial mathematics, conducted through human-directed, AI-assisted research. It presents finite computational results and does not claim a proof of an infinite family of consecutive-emirp magic squares.