On the largest prime factor of integers in short intervals III
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908744769601536 |
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| author | Li, Runbo |
| author_facet | Li, Runbo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00910 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the largest prime factor of integers in short intervals III Li, Runbo Number Theory 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. |
| title | On the largest prime factor of integers in short intervals III |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.00910 |