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

Harman, G.

doi:10.1090/S0025-5718-05-01749-7

On the greatest prime factor of $p-1$ with effective constants
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Math. Comp. 74 (2005), 2035-2041 Request permission

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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