2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132835Let $f\in \mathbb{Z}\lbrack x\rbrack$ be a polynomial of degree $d\geq 3$ without roots of multiplicity $d$ or $(d-1)$. Erdős conjectured that, if $f$ satisfies the necessary local conditions, then $f(p)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov's theorem from the theory of large deviations.39 pages; rather major revision, with strengthened and generalized statementsNumber Theory11N32; 11D45, 11G05, 11G30, 11N25text