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arXiv · 1605.05568

On a Diophantine inequality involving a prime and an almost-prime

Liyang Yang
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Abstract

We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r| \frac{λ_1}{λ_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$

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Liyang Yang. 2016-05-20. On a Diophantine inequality involving a prime and an almost-prime. https://arxiv.org/abs/1605.05568

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