Skip to content
DraconiticResearch
Search Sources API
194,161 studies31 sourcesUpdated 42 min ago
PreprintarXiv licencearXiv

An improvement on the largest prime factor of n2+1

  • Hector Pasten
Open at sourcePDFDOICite

Abstract

The study of the largest prime factor in polynomial sequences can be traced back at least to the late 19th century in the work of Störmer. Mahler (1933) and Chowla (1934) proved that the largest prime factor of n2+1 grows at least as fast as log2 n. In 2023 we improved this to (log2 n)2/log3 n. In this note we show the lower bound (log2 n)2/log4 n, and that when this bound is nearly sharp it also holds for many prime factors of n2+1.

Related

  1. On the largest prime factors of p−1 and p+1
    Genheng Zhao
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Rigidity of Averages over the Two Largest Prime Factors
    Dijia Chen
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Note on the largest prime factors of consecutive integers
    Zhiyuan Yang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. On the largest prime factors less than y of consecutive shifted primes
    Zhiyuan Yang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields
    Yuji Hashimoto, Koji Nuida
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{pasten2026improvement,
  title = {{An improvement on the largest prime factor of $n^2+1$}},
  author = {Hector Pasten},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01327},
  eprint = {2609.01327},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01327},
}
RIS
TY  - UNPB
TI  - An improvement on the largest prime factor of n^2+1
AU  - Hector Pasten
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01327
UR  - https://arxiv.org/abs/2609.01327
AB  - The study of the largest prime factor in polynomial sequences can be traced back at least to the late 19th century in the work of Störmer. Mahler (1933) and Chowla (1934) proved that the largest prime factor of n^2+1 grows at least as fast as log_2 n. In 2023 we improved this to (log_2 n)^2/log_3 n. In this note we show the lower bound (log_2 n)^2/log_4 n, and that when this bound is nearly sharp it also holds for many prime factors of n^2+1.
ER  -