Solution to a problem of Luca, Menares and Pizarro-Madariaga

Let $k\ge 2$ be a positive integer and $P^{+}(n)$ the greatest prime factor of a positive integer $n$ with convention $P^{+}(1)=1$. For any $\theta \in [\frac{1}{2k},\frac{17}{32k})$, set $$T_{k,\theta }(x)=\sum _{\begin{array}{c}{p}_1\cdot \cdot \cdot p_k\le x\\ P^{+}(\gcd (p_1-1,\mathrm{.}\mathrm{.}\mathrm{.},p_k-1))\ge (p_1\cdot \cdot \cdot p_k{)}^{\theta }\end{array}}1,$$ where the $p^{\mathrm{\prime }}$s are primes. It is proved that $$T_{k,\theta }(x){\ll }_k\frac{x^{1-\theta (k-1)}}{(\log x{)}^2},$$ which, together with the lower bound $$T_{k,\theta }(x){\gg }_k\frac{x^{1-\theta (k-1)}}{(\log x{)}^2}$$ obtained by Wu in 2019, answer a 2015 problem of Luca, Menares and Pizarro-Madariaga on the exact order of magnitude of $T_{k,\theta }(x)$. A main novelty in the proof is that, instead of using the Brun–Titchmarsh theorem to estimate the $k^{th}$ movement of primes in arithmetic progressions, we transform the movement to an estimation involving taking primes simultaneously by linear shifts of primes.