On a ternary Diophantine problem with mixed powers of primes

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 that ${\lambda }_1/{\lambda }_2$ is irrational and $\varpi$ is any real number, then for any $\eps > 0$ the inequality $ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^k + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-(33 - 29 k) / (72 k) + \eps} $ has infinitely many solutions in prime variables $p_1$, ..., $p_k$.

arXiv commentrevised version with new title of 1206.0250; submitted. arXiv admin note: substantial text overlap with arXiv:1206.0252, arXiv:1206.0246

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