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On the greatest prime factor of $p-1$ with effective constants

Author: G. Harman
Journal: Math. Comp. 74 (2005), 2035-2041
MSC (2000): Primary 11N13
Published electronically: February 16, 2005
MathSciNet review: 2164111
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Abstract: Let $p$ denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of $p-1$ exceeding $(p-1)^{\frac 12}$ in which the constants are effectively computable. As a result we prove that it is possible to calculate a value $x_0$ such that for every $x > x_0$ there is a $p < x$ with the greatest prime factor of $p-1$ exceeding $x^{\frac 35}$. The novelty of our approach is the avoidance of any appeal to Siegel’s Theorem on primes in arithmetic progression.

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Additional Information

G. Harman
Affiliation: Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom

Received by editor(s): March 19, 2004
Received by editor(s) in revised form: August 16, 2004
Published electronically: February 16, 2005
Article copyright: © Copyright 2005 American Mathematical Society