On the largest prime factor of integers in short intervals III
Abstract
Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently large $x$. This gives a solution with $γ= \frac{1}{36}$ to the Exercise 5.1 in Harman's monograph and improves the previous record of the author proved in 2024, where $γ= \frac{1}{26.5}$ is obtained.
Fields of Study & Subjects
Analytic Number Theory Research
Limits and Structures in Graph Theory
Advanced Mathematical Identities
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 1 January 2026
- Document type
- Preprint