2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/136169We give a sufficient and necessary condition for a p-adic integer to have p-th root in the ring of p-adic integers. The same condition holds clearly for residues modulo p^k. We give a proof that Fermat's last theorem is false for p-adic integers and for residues mod p^k.7 pages, no figure, submittedNumber Theory11S99; 13K05text