2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149225We consider the (A_n,A_1) Y-system arising in the Thermodynamic Bethe Ansatz. We prove the periodicity of solutions of this Y-system conjectured by Al.B. Zamolodchikov, and the dilogarithm identities conjectured by F. Gliozzi and R. Tateo.15 pages, AMSLATEXHigh Energy Physics - TheoryAlgebraic Geometrytext