# Misunderstanding Euclid's argument

**Number Theory**Level 2

Assume there are only a finite number of primes \(p_1,p_2,\ldots,p_n\). Let \(N\) be the product of all of those primes, add to it \(1\) and you get a

newprime number since it isn't divisible by any of the primes we listed at first. Contradiction! \(\Rightarrow\Leftarrow\) Therefore, there is an infinite number of primes.

Let \( q_1, q_2, q_3, \ldots \) be the list of all primes (in ascending order). Your mission is to find the smallest value of \( q_1 q_2 q_3 \ldots q_n + 1 \) that is not a prime.

**Your answer seems reasonable.**Find out if you're right!

**That seems reasonable.**Find out if you're right!

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