On a Family of Congruent Numbers Defined Modulo 72: A Conjectural Study

We investigate the interplay between the Chinese Remainder Theorem and the theory of congruent numbers through a modular approach to expressing integers as sums of three cubes. By analyzing congruence systems arising from specific residue classes modulo 8 and modulo 9, we classify the possible integers likely to be representable as sums of cubes based on their modular residues. We also explore computational methods applying this modular framework to identify explicit cube decompositions within these classes.