The largest prime factor of n2+1and improvements on subexponential ABC

• 成果类型: Article; Early Access
• 署名作者: Pasten, Hector
• 署名单位: Pontificia Universidad Catolica de Chile
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01244-6
• 发表日期: 2024
• 关键词: elliptic-curves

摘要

We combine transcendental methods and the modular approaches to the ABC conjecture to show that the largest prime factor of n(2)+1isatleastofsize(log(2)n)(2)/log(3)nwhere log k is the k-th iterate of the logarithm. This gives a substantial improvement on the best available estimates, which are essentially of size log2ngoing back to work of Chowla in 1934. Using the same ideas, we also obtain significant progress on sub expoential bounds for the ABC conjecture, which in a case gives the first improvement on a result by Stewart and Yu dating back over two decades. Central toour approach is the connection between Shimura curves and the ABC conjecture developed by the author.