a step beyond Euclid and Fermat, 1

The eternally elegant Euclid's proof of infinitude of the set of prime numbers can be rephrased as follows: let

• e(0)  :=  2
  
• e(n+1)  :=  (e(n) - 1) ⋅ e(n) + 1

Then  (e(n) : n=1 2...)  is an increasing sequence of natural numbers such that each of its two different terms are relatively prime.

Indeed,

• e(n+1)  :=  1 + ∏_k=1...n e(k)

for every  n = 0 1 ...

Let  (p(k) : k=0 1 ...)  be the increasing sequence of all prime numbers. We see that

• p(n)  ≤  e(n)   for every n=0 1...

In fact:

• p(0) = e(0) = 2
• p(1) = e(1) = 3

but

• p(2) = 5 < 7 = e(2)

and

• p(n)  <  e(n)   for every n=2 3...