Heath-Brown, D. R. (2001) *The largest prime factor of .* Proceedings of the London Mathematical Society (3), 82 . pp. 554-596.

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## Abstract

The largest prime factor of has been investigated by Hooley, who gave a conditional proof that it is infinitely often at least as large as , with a certain positive constant . It is trivial to obtain such a result with . One may think of Hooley's result as an approximation to the conjecture that is infinitely often prime. The condition required by Hooley, his R conjecture, gives a non-trivial bound for short Ramanujan-Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of is infinitely often at least as large as , though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non-trivial bound for short Ramanujan-Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus.

Item Type: | Article |
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Additional Information: | This is a pre-print version. The original journal should be consulted for the final version. |

Uncontrolled Keywords: | Largest prime factor, Cubic polynomial, Hypothesis R$^{*}$, Unconditional |

Subjects: | H - N > Number theory |

Research Groups: | Number Theory Group |

ID Code: | 165 |

Deposited By: | Roger Heath-Brown |

Deposited On: | 27 Jan 2005 |

Last Modified: | 20 Jul 2009 14:19 |

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