Completing Diophantus, De polygonis numeris, prop. 5 (Q643313)

In his \textit{De polygonis numeris}, Diophantus enunciates a statement, prop.\ 5, asking ``Given a number, to find in how many ways it can be polygonal.'' Since the text of prop.\ 5 breaks off, there have been several attempts to either complete it ([\textit{G. Wertheim}, Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Leipzig: B. G. Teubner (1890; JFM 22.0011.02)] and [\textit{T. L. Heath}, \textit{Diophantus} of Alexandria, a study in the history of Greek algebra. Cambridge: University Press (1910; JFM 41.0003.02)]), or answer the question without attempting to continue with the proof where it breaks off [Bachet (1621)], or to explain it away as ``a worthless attempt of a commentator'' [\textit{P. Tannery}, Diophanti Alexandrini opera omnia cum graecis commentariis. Edidit et latine interpretatus est P. Tannery. Vol. I. Lipsiae: B. G. Teubner (1893; JFM 25.0014.01)]. The author finds the reconstructions unsatisfactory and proposes his own, one that is more ``Greek looking'' than its predecessors.