A conjecture of Erdos, supersingular primes and short character sums

• 成果类型: Article
• 署名作者: Bennett, Michael A.; Siksek, Samir
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.191.2.2
• 发表日期: 2020
• 页码: 355-392
• 关键词: diophantine equations; perfect powers; modular-representations; consecutive terms; rational-points; elliptic-curves; PRODUCTS

摘要

If k is a sufficiently large positive integer, we show that the Diophantine equation n(n + d) center dot center dot center dot (n + (k-1)d) = y(l) has at most finitely many solutions in positive integers n, d, y and, 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.