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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

Online
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1435-5345
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Volume 2012, Issue 667

Issues

Almost prime Pythagorean triples in thin orbits

Alex Kontorovich / Hee Oh
Published Online: 2011-08-12 | DOI: https://doi.org/10.1515/CRELLE.2011.128

Abstract

For the ternary quadratic form Q(x) = x2 + y2 − z2 and a non-zero Pythagorean triple x0 ∈ ℤ3 lying on the cone Q(x) = 0, we consider an orbit 𝒪 = x0Γ of a finitely generated subgroup Γ < SOQ(ℤ) with critical exponent exceeding 1/2.

We find infinitely many Pythagorean triples in 𝒪 whose hypotenuse, area, and product of side lengths have few prime factors, where “few” is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants.

About the article

Received: 2010-01-05

Revised: 2010-11-07

Published Online: 2011-08-12

Published in Print: 2012-06-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2012, Issue 667, Pages 89–131, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2011.128.

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[1]
Jean Bourgain and Alex Kontorovich
International Mathematics Research Notices, 2015, Volume 2015, Number 19, Page 9175
[2]
Min Lee and Hee Oh
Geometric and Functional Analysis, 2013, Volume 23, Number 2, Page 580
[3]

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