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.
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Acknowledgements
I thank Kálmán Györy and Cameron L. Stewart for valuable comments on these results. And I am particularly indebted to Samuel Le Fourn and M. Ram Murty for carefully reading an earlier version of this manuscript and suggesting several changes and corrections. The comments and suggestions of the referee are gratefully acknowledged.
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Supported by ANID Fondecyt Regular grant 1230507 from Chile.
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Dedicado a la memoria de mi padre, quien siempre me apoyó en todo
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Pasten, H. The largest prime factor of \(n^{2}+1\) and improvements on subexponential \(ABC\). Invent. math. 236, 373–385 (2024). https://doi.org/10.1007/s00222-024-01244-6
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DOI: https://doi.org/10.1007/s00222-024-01244-6