On a ternary Diophantine problem with mixed powers of primes
Volume 159 / 2013
Acta Arithmetica 159 (2013), 345-362
MSC: Primary 11D75; Secondary 11P55, 11N05.
DOI: 10.4064/aa159-4-4
Abstract
Let $1 < k < 33 / 29$. We prove that if $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real numbers, not all of the same sign and such that $\lambda_1 / \lambda_2$ is irrational, and $\varpi$ is any real number, then for any $\varepsilon > 0$ the inequality $\vert\lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^k +\varpi\vert \le( \max_j p_j )^{-(33 - 29 k) / (72 k) + \varepsilon}$ has infinitely many solutions in prime variables $p_1$, $p_2$, $p_3$.
