The Mathematical Institute, University of Oxford, Eprints Archive

The largest prime factor of $X^3+2$

Heath-Brown, D. R. (2001) The largest prime factor of $X^3+2$. Proceedings of the London Mathematical Society (3), 82 . pp. 554-596.

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Abstract

The largest prime factor of $X^3+2$ has been investigated by Hooley, who gave a conditional proof that it is infinitely often at least as large as $X^{1+\delta}$, with a certain positive constant $\delta$. It is trivial to obtain such a result with $\delta=0$. One may think of Hooley's result as an approximation to the conjecture that $X^3+2$ 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 $X^3+2$ is infinitely often at least as large as $X^{1+\delta}$, 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
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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