On the largest prime factor of

  • Jori Merikoski

    University of Oxford, UK
On the largest prime factor of $n^2+1$ cover
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We show that the largest prime factor of is infinitely often greater than . This improves the result of de la Bretèche and Drappeau (2020) who obtained this with in place of The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman’s sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent may be increased to

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Jori Merikoski, On the largest prime factor of . J. Eur. Math. Soc. 25 (2023), no. 4, pp. 1253–1284

DOI 10.4171/JEMS/1216