2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116442In this paper we introduce the notion of the $P$-sequences and apply their properties in studying representability of real numbers. Another application of $P$-sequences we find in generating the Prouhet-Tarry-Escott pairs.Number Theory11D41; 11B05;11B75;11Y50Sums of Like Powers and Some Dense Setstext