# 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

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

- Infinitude of Primes
- Infinitude of Primes - A Topological Proof
- Infinitude of Primes - A Topological Proof without Topology
- Infinitude of Primes Via *-Sets
- Infinitude of Primes Via Coprime Pairs
- Infinitude of Primes Via Fermat Numbers
- Infinitude of Primes Via Harmonic Series
- Infinitude of Primes Via Lower Bounds
- Infinitude of Primes - via Fibonacci Numbers
- New Proof of Euclid's Theorem
- Infinitude Of Primes From Legendre's Formula
- Infinitude of Primes - via Bertrand's Postulate
- Infinitude of Primes - An Impossible Injection
- Infinitude of Primes - from Infinitude of Mutual Primes
- Infinitude of Primes Via Euler's Product Formula
- Infinitude of Primes Via Euler's Product Formula for Pi
- Infinitude of Primes Via Powers of 2
- Infinitude of Primes As a Quickie
- A One-Line Proof of the Infinitude of Primes
- Infinitude of Primes - A. Thue's Proof
- Why The Number of Primes Could Not Be Finite?

|Contact| |Front page| |Contents| |Algebra| |Up|

Copyright © 1996-2018 Alexander Bogomolny

66156645