Since \(N > 1\), the fundamental theorem gives it a prime factorization; that prime must be in the list by hypothesis, yet it divides both \(N\) and \(N-1\), hence it divides 1. The option saying \(N\) is less than 1 is false, since \(N > 1\).
The infinitude of primes is Euclid's theorem stating that the set of prime numbers is infinite: given any finite list of primes \(p_1,\dots,p_n\), there is a prime not on it.
\[p\mid p_1p_2\cdots p_n+1\ \Longrightarrow\ p\neq p_i\ \text{for all } i\]