On a conjecture of Erdős (Q5918497)

The authors show that for any given integers \(a_1 < a_2 < \cdots < a_k\) there are infinitely integers \(n>0\) such that the number of representations \(n = p+a_i\) with prime \(p\) exceeds \[ \frac{\log k}8-1.6. \] This implies the truth of the following conjecture posed in [\textit{P. Erdős}, Summa Brasil. Math. 2, 113--123 (1950; Zbl 0041.36808)]: If \(c>0\) and integers \(a_1<a_2<\cdots<a_t\le x\) are given with \(t>\log x\) and \(x\) is sufficiently large, then the number of solutions of \(n=p+a_i\) with prime \(p\) exceeds \(c\).