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

2016/05/18 by Liyang Yang, Yang, Liyang
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1605.05568

openalex publication_date 2016/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that there are infinitely many solutions of |λ₀+λ₁p+λ₂P_r|(λ₁)/(λ₂) not in ℚ. This improves a result by Harman. Moreover, we show that one can require the prime p to be of the form \floorn^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 (1, 1+\frac1149].

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