Question 47
Given any positive integer $k$, prove that there exist a prime $p$ and distinct integers $a_1,a_2,\ldots,a_{k+3}\in\{1,2,\ldots,p-1\}$ such that
$$
p\mid a_i a_{i+1}a_{i+2}a_{i+3}-i
\qquad(1\le i\le k).
$$
$$
p\mid a_i a_{i+1}a_{i+2}a_{i+3}-i
\qquad(1\le i\le k).
$$