On Diophantine problems with mixed powers of primes
Volume 182 / 2018
Abstract
Let be an integer with k\geq 3 and \varepsilon \gt 0. Let s(k)=[{(k+1)}/{2}] and \sigma(k)=\min\bigl(2^{s(k)-1},\frac{1}{2}s(k)(s(k)+1)\bigr). Suppose that \lambda_1,\lambda_2,\lambda_3 are non-zero real numbers, not all negative, and \lambda_1/\lambda_2 is irrational and algebraic. Let \mathcal{V} be a well-spaced sequence and \delta \gt 0. We prove that E(\mathcal{V},X,\delta)\ll X^{1-{1}/{(8\sigma(k))}+2\delta+\varepsilon}, where E(\mathcal{V},X,\delta) denotes the number of v\in \mathcal{V} with 1\leq v\leq X such that the inequality |\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^k-v| \lt v^{-\delta} has no solution in primes p_1,p_2,p_3. Furthermore, suppose that \lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5 are non-zero real numbers, not all of the same sign, \lambda_1/\lambda_2 is irrational and \varpi is a real number. We prove that there are infinitely many solutions in primes p_j to the inequality |\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\lambda_5p_5^k+\varpi| \lt (\max p_j)^{-{1}/{(8\sigma(k))}+\varepsilon}. This gives an improvement of an earlier result.