On the Largest Prime Factor of Integers in Short Intervals

The author sharpens a result of Jia and Liu (2000), showing that for sufficiently large \( x \), the interval \( [x, x+x^{\frac{1}{2}+\varepsilon}] \) contains an integer with a prime factor larger than \( x^{\frac{51}{53}-\varepsilon} \). This gives a solution with \( \gamma = \frac{2}{53} \) to the Exercise 5.1 in Harman's monograph.