# Prove that if $R$ is not prime then $R$ must have a prime factor $q$ that is larger than $p_n$.

This is a modification of my previous post here which has completely different meanings/solutions

Given that

$$R = p_1p_2\cdots p_n + 1$$ where $$p_1 < p_2 < \cdots < p_n$$ and $$p$$ are the first n prime numbers.

Prove that if $$R$$ is not prime then $$R$$ must have a prime factor $$q$$ that is larger than $$p_n$$.

I directly understand that this question refers to Euclid's primes proof however; I don't know really how to even tackle this problem. I am looking over euclid's proof and will hopefully run into a 'eureka' moment.

Any advice or tips on how to solve this problem would be very helpful and appreciated.

• This was answered in your prior question. You know from that answer that the prime factors of $R$ are not among the first n primes, therefore they are all larger than the nth prime. Feb 14, 2014 at 0:32
• Sorry, I may have missed it!
– A A
Feb 14, 2014 at 0:44

Hint: If $R$ is not prime, then $R$ has a prime divisor. Convince yourself that this prime divisor isn't any of $p_1, ..., p_n$. Now use how $p_1, ..., p_n$ are defined.