On a theorem of Prachar involving prime powers (Q2898389)

The authors of this paper use the circle method to investigate the solubility of the Waring-Goldbach equation NEWLINE\[NEWLINEn = p_2^2 + p_3^3 + p_4^4 + p_5^5,NEWLINE\]NEWLINE where \(n\) is an even positive integer and the \(p_i\)'s are odd primes. Let \(N\) and \(U\) be large numbers and let \(E(N, U)\) denote the number of \(n\) in the interval \([N, N+U]\) such that the above equation has no solution satisfying \(|p_k^k -N/4| \leq U\) for \(k =2, 3, 4, 5\). Theorem 1 of the paper says that for \(U = N^{1 -\frac{1}{36} + \varepsilon}, \) it is true that \(E(N, U) \ll N^{1 -\varepsilon}\). In their notation, \(\varepsilon >0\) is arbitrary. NEWLINENEWLINETheorem 3 in the paper says that for any sufficiently large odd integer \(N\), the equation NEWLINE\[NEWLINEN = p_1 + p_2^2 + p_3^3 +p_4^4 +p_5^5NEWLINE\]NEWLINE has solutions with \(|p_k^k -\frac{N}{5}| \leq U, (k =1, 2, 3, 4, 5\)) for \(U = N^{1-\frac{1}{264} +\varepsilon}\).