# Infinitude of Primes via Coprime Pairs

For any integer $n \gt 1$, $n$ and $n+1$ are coprime - mutually prime, having no common prime factors. So start with any $n \gt 1$ and write down one of its prime factors, say $p$. The prime factors of its successor, $n + 1$, are different from $p$. So there is at least some other prime, say $q$.

Now consider the successor of the product $n(n + 1)$. The prime factors of the latter are different from those of n and $n + 1$, $p$ and $q$, in particular. Let $r$ be one of those.

Appply the same argument to the successor of $n(n + 1)[n(n + 1) + 1]$ to obtain yet another prime, say $s$. Obviously the process can be extended indefinitely.

(I am grateful to Jens Bossaert for pointing out the source of the proof. The proof is due to Filip Saidak, a mathematician at the University of North Carolina. Much later I found in Victor Moll's book a generalization that I placed on a separate page.)

## Reference

1. V. H. Moll, Numbers and Functions: From a Classical-Experimental Mathematician's Point of View, AMS, 2012, 26-27
2. F. Saidak, A New Proof of Euclid's Theorem, The American Mathematical Monthly, Vol. 113, No. 10 (Dec., 2006), 937-938