I was reading this paper: http://fermatslibrary.com/s/a-new-proof-of-euclids-theorem and became confused when reading this line:
Since $n$ and $n + 1$ are consecutive integers, they must be coprime. Hence the number $N_2 = n(n + 1)$ must have at least two diﬀerent prime factors
I see that this is true in practice when writing out several examples, but was hoping for an explanation of why? (which the paper left out). I understand why $n$ and $n+1$ are co-prime, but not why that implies that $n\cdot(n+1)$ has at least two different prime factors. What does being co-prime have to do with this?
Any insight would be really helpful.
I believe I understand now:
1) $n$, $n+1$ have no factors in common except $1$
2) If $n$, $n+1 \ge 2$ they have at least one prime factor
3) From (1) these factors must be different, so there must be at least two different prime factors