Small prime solutions to diagonal Diophantine equations.

Let k ≥ 3 be an integer, s = 2 k + 1 , (a i , a j) = 1 , 1 ≤ i < j ≤ s , where a 1 , ... , a s are nonzero integers, and let n be an integer. Suppose that a 1 , ... , a s satisfy some necessary congruent conditions. In this paper, it is proved that (i) if a j are not all of the same sign, then the equation a 1 p 1 k + ⋯ + a s p s k = n has prime solutions satisfying p j ≪ | n | 1 / k + max { | a j | } C (k) + 𝜀 , (ii) if all a j are positive and n ≫ max { | a j | } k C (k) + 1 + 𝜀 , then a 1 p 1 k + ⋯ + a s p s k = n is solvable in primes p j , where C (k) = 1 + 6 ⋅ 2 k + 2 0 3 ⋅ 2 2 k − 1 + 3 ⋅ 2 k − 1 − 2 0 . Our result uses C (3) = 3 9 2 2 ≈ 1. 7 7 2 7 ⋯ as an improvement of the recent result C (3) = 2 due to Zhao (2016), and largely improves the results C (k) = 3 ⋅ 2 k − 1 for k ≥ 4 proved by Yang and Hu (2016). [ABSTRACT FROM AUTHOR]