On the largest prime factor of quadratic polynomials

Runbo Li

On the largest prime factor of quadratic polynomials

Number Theory 2025-06-03 v2

Authors: Runbo Li

Abstract

Let $x$ denote a sufficiently large integer. We show that the recent result of Grimmelt and Merikoski actually yields the largest prime factor of $n^2 +1$ is greater than $x^{1.317}$ infinitely often. As an application, we give a new upper bound for the number of integers $n \leqslant x$ which $n^2 +1$ has a primitive divisor.

Keywords

prime number theory prime numbers perfect numbers

Cite

@article{arxiv.2406.07575,
  title  = {On the largest prime factor of quadratic polynomials},
  author = {Runbo Li},
  journal= {arXiv preprint arXiv:2406.07575},
  year   = {2025}
}

Comments

5 pages. Version 1 is another preprint which gives a weaker result

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