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Triangles with prime hypotenuse
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Chow, Sam and Pomerance, Carl (2017) Triangles with prime hypotenuse. Research in Number Theory, 3 (21). doi:10.1007/s40993-017-0086-6 ISSN 2363-9555.
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Official URL: https://doi.org/10.1007/s40993-017-0086-6
Abstract
The sequence 3,5,9,11,15,19,21,25,29,35,… consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erdős–Ford–Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes in narrow sectors, and the Hardy–Ramanujan inequality.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Gaussian processes, Pythagorean theorem | ||||||||
Journal or Publication Title: | Research in Number Theory | ||||||||
Publisher: | SpringerOpen | ||||||||
ISSN: | 2363-9555 | ||||||||
Official Date: | December 2017 | ||||||||
Dates: |
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Volume: | 3 | ||||||||
Number: | 21 | ||||||||
DOI: | 10.1007/s40993-017-0086-6 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 19 September 2019 | ||||||||
Date of first compliant Open Access: | 19 September 2019 | ||||||||
RIOXX Funder/Project Grant: |
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