Note on shifted primes with large prime factors

Yuchen Ding, Zhiwei Wang

We denote by $P^{+}(n)$ the largest prime factor of the integer $n$. In 1935, Erd\H os studied the quantity $T_c(x)$ defined by $$ T_c(x)=\big|\big\{p\le x: P^+(p-1)\ge p^c\big\}\big…