Journal article
Triangles with prime hypotenuse
- 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.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Version of record, pdf, 634.8KB, Terms of use)
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- Publisher copy:
- 10.1007/s40993-017-0086-6
Authors
- Publisher:
- Springer Open
- Journal:
- Research in Number Theory More from this journal
- Volume:
- 3
- Issue:
- 1
- Article number:
- 21
- Publication date:
- 2017-10-09
- Acceptance date:
- 2017-06-27
- DOI:
- ISSN:
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2363-9555
- Keywords:
- Pubs id:
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pubs:926124
- UUID:
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uuid:24e35e17-405e-40cd-bfec-85052f1b1c88
- Local pid:
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pubs:926124
- Source identifiers:
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926124
- Deposit date:
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2018-10-10
Terms of use
- Copyright holder:
- Chow and Pomerance
- Copyright date:
- 2017
- Notes:
- Copyright © 2017 The Authors. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
- Licence:
- CC Attribution (CC BY)
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