On the largest prime factor of shifted primes.

For any integer n ≥ 2, let P( n) be the largest prime factor of n. In this paper, we prove that the number of primes p ≤ x with P( p−1) ≥ p is more than (1− c+ o(1)) π( x) for 0 < c < 1/2. This extends a recent result of Luca, Menares and Madariaga for 1/4 ≤ c ≤ 1/2. We also pose two conjectures for further research. [ABSTRACT FROM AUTHOR]