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

2023/12/06 by Héctor Pastén, Pasten, Hector
Arts and Humanities · Mathematics · #11G18 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT) #Primary: 11J25 #Secondary: 11J86

paper · pdf · doi:10.48550/arxiv.2312.03566

openalex publication_date 2023/12/06 · openalex created_date 2023/12/08 · openalex updated_date 2026/07/28

Abstract

We combine transcendental methods and the modular approaches to the ABC conjecture to show that the largest prime factor of n²+1 is at least of size (log₂ n)²/log₃n where 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 log₂ n going back to work of Chowla in 1934. Using the same ideas, we also obtain significant progress on subexpoential 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 to our approach is the connection between Shimura curves and the ABC conjecture developed by the author.

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