Representations of large integers as the sum of fractional powers of primes and squares

Conjecture H of Hardy and Littlewood says that every large integer is either a square or the sum of a prime and a square. Despite progress showing that this is true for almost all large integers, Conjecture H remains unproven. In this paper, Conjecture H is approximated using Piatetski–Shapiro sequences by showing that all large integers can be written in the form [Formula: see text] for any fixed [Formula: see text]. The existing results for the conjecture are also extended by showing that almost all large integers can be written in the form [Formula: see text] for any fixed [Formula: see text].