Small prime solutions to diagonal Diophantine equations

Abstract

Let [Formula: see text] be an integer, [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] are nonzero integers, and let [Formula: see text] be an integer. Suppose that [Formula: see text] satisfy some necessary congruent conditions. In this paper, it is proved that (i) if [Formula: see text] are not all of the same sign, then the equation [Formula: see text] has prime solutions satisfying [Formula: see text], (ii) if all [Formula: see text] are positive and [Formula: see text], then [Formula: see text] is solvable in primes [Formula: see text], where [Formula: see text]. Our result uses [Formula: see text] as an improvement of the recent result [Formula: see text] due to Zhao (2016), and largely improves the results [Formula: see text] for [Formula: see text] proved by Yang and Hu (2016).