THE INTEGER PARTS OF A NONLINEAR FORM WITH MIXED POWERS 3 AND k

Abstract

Using the Davenport-Heilbronn circle method, we show that if $\lambda_1,\cdots,\lambda_5$ are positive real numbers, at least one of the ratios $\lambda_i/\lambda_j(1\leq i\lt j\leq 5)$ is irrational, then, for arbitrary positive integer $k\geq 4$, the integer parts of $\lambda_1 x_1^3 + \lambda_2 x_2^3 + \lambda_3 x_3^3 + \lambda_4 x_4^3 +\lambda_5 x_5^k$ are prime infinitely often for natural numbers $x_1,\cdots,x_5$.