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 $\theta\in \left[\frac 1{2k},\frac{17}{32k}\right)$, set $$T_{k,\theta}(x)=\sum_{\substack{p_1\cdot\cdot\cdot p_k\le x\\ P^+(\gcd(p_1-1,...,p_k-1))\ge (p_1\cdot\cdot\cdot p_k)^\theta}}1,$$ where the $p'$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.