Combinatorial proofs of five formulas of Liouville

Liouville gave formulas for the number of representations of a positive integer by the quaternary quadratic forms x 2 + y 2 + 2 z 2 + 2 t 2 , x 2 + y 2 + z 2 + 4 t 2 , x 2 + y 2 + 4 z 2 + 4 t 2 , x 2 + 4 y 2 + 4 z 2 + 4 t 2 and x 2 + 2 y 2 + 2 z 2 + 4 t 2 . These formulas have been proved by a number of authors by a variety of non-elementary methods. We give combinatorial proofs of these formulas by deducing them from Jacobi’s four squares theorem and Legendre’s four triangular numbers theorem. Since these latter two theorems have both been proved in an elementary arithmetic way, the five formulas of Liouville are therefore proved in a completely elementary way.