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A conjecture of Erdős, supersingular primes and short character sums

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Siksek, Samir and Bennett, Michael A. (2020) A conjecture of Erdős, supersingular primes and short character sums. Annals of Mathematics, 191 (2). pp. 355-392. doi:10.4007/annals.2020.191.2.2 ISSN 0003-486X.

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Official URL: https://doi.org/10.4007/annals.2020.191.2.2

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Abstract

If k is a sufficiently large positive integer, we show that the Diophantine equation
n(n+d)⋯(n+(k−1)d)=yℓ
has at most finitely many solutions in positive integers n,d,y and ℓ, with gcd(n,d)=1 and ℓ≥2. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Curves, Elliptic, Galois theory, Modular functions, Diophantine equations, Number theory
Journal or Publication Title: Annals of Mathematics
Publisher: Mathematical Sciences Publishers ; Princeton University
ISSN: 0003-486X
Official Date: 13 February 2020
Dates:
DateEvent
2020UNSPECIFIED
13 February 2020Available
12 November 2019Accepted
Volume: 191
Number: 2
Page Range: pp. 355-392
DOI: 10.4007/annals.2020.191.2.2
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 27 November 2019
Date of first compliant Open Access: 27 November 2019
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIED[NSERC] Natural Sciences and Engineering Research Council of Canadahttp://dx.doi.org/10.13039/501100000038
EP/K034383/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
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