An elementary proof is given of an arithmetic formula, which was stated but not proved by Liouville. An application of this formula yields a formula for the number of representations of a positive integer as the sum of twelve triangular numbers.

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Journal Journal de Theorie des Nombres de Bordeaux
McAfee, E. (Erin), & Williams, K.S. (2006). An arithmetic formula of Liouville. Journal de Theorie des Nombres de Bordeaux, 18(1), 223–239. doi:10.5802/jtnb.541