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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
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 | |||||||||
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| 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: |
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| 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: |
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