Let us define the sequence of primes in the following way: $p_1 = 2$, and if $p_1,p_2...p_k$ are given, then let $p_{k+1}$ be the largest prime divisor of $p_1p_2...p_k + 1$ (so $p_2=3, p_3=7, etc.)$. Show that 5 is not in this sequence.
This is one of the challenging questions from my book, which is fitting because I honestly have no idea about how to do this. If anyone could solve this and explain it, it would be a huge help. Many thanks in advance.
Once you have $p_1=2,p_2=3,p_3=7$ the only way $5$ can appear is if $p_1p_2p_3\ldots p_k+1$ is a power of $5$. Otherwise there will be some higher prime factor. $5^m \equiv 1 \pmod 4$ but the product of primes is equivalent to $2 \pmod 4$ so the product of primes plus one is not a power of $5$.