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

Let $p$ denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of $p-1$ exceeding $(p-1)^{\frac12}$ 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 $x^{\frac35}$. The novelty of our approach is the avoidance of any appeal to Siegel's Theorem on primes in arithmetic progression.