The largest prime factor of $n^{2}+1$ and improvements on subexponential $ABC$

Abstract

We combine transcendental methods and the modular approaches to the \(ABC\) conjecture to show that the largest prime factor of \(n^{2}+1\) is at least of size \((\log _{2} n)^{2}/\log _{3}n\) 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 \(\log _{2} n\) 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.