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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.

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    $\begingroup$ 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. $\endgroup$
    – Bill Dubuque
    Feb 14, 2014 at 0:32
  • $\begingroup$ Sorry, I may have missed it! $\endgroup$
    – A A
    Feb 14, 2014 at 0:44

1 Answer 1

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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.

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  • $\begingroup$ Thank you for the response Tyler. I think I have figured it out! 1. q cannot be in the given set p 2. q is a prime divisor of R 3. Since q is not in the set p, which are the first primes in order, q must be greater than the greatest number in that list. $\endgroup$
    – A A
    Feb 14, 2014 at 0:36
  • $\begingroup$ @AA Exactly right. $\endgroup$
    – user61527
    Feb 14, 2014 at 0:36

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