MathNet 1987 — Part 01 Question 24

Define the sequence $\{p\}_1, \{p\}_2, \{p\}_3, \ldots$ as follows. $\{p\}_1 = 2$, and $\{p\}_n$ is the largest prime divisor of $\{p\}_1 \{p\}_2 \ldots \{p\}_{n-1} + 1$. Prove that $5$ does not occur in the sequence.