# Misunderstanding Euclid's argument

A common flawed presentation of Euclid's proof of the infinitude of primes is as follows:

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 new prime 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.

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