On the largest prime factors of consecutive integers in short intervals

For an integer n > 1, let P(n) be the largest prime factor of n. We prove that, for x → ∞, there exists a positive proportion of consecutive integers n and n + 1 such that P(n) < P(n + 1) in short intervals (x, x + y] with x7/12 < y ⩽ x. In particular, we have | { n ⩽ x : P ( n ) < P ( n...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2017-08, Vol.145 (8), p.3211-3220
Main Author: WANG, ZHIWEI
Format: Article
Language:English
Subjects:
A. ALGEBRA, NUMBER THEORY, AND COMBINATORICS
Mathematics
Number Theory
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Summary:For an integer n > 1, let P(n) be the largest prime factor of n. We prove that, for x → ∞, there exists a positive proportion of consecutive integers n and n + 1 such that P(n) < P(n + 1) in short intervals (x, x + y] with x7/12 < y ⩽ x. In particular, we have | { n ⩽ x : P ( n ) < P ( n + 1 ) } | ≥ 0.1063 x . This improves a previous result of La Bretèche, Pomerance and Tenenbaum.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/13459