a step beyond Euclid and Fermat, 2

Fermat has defined his numbers:

• f(n) := 2²ⁿ + 1 for every n=0 1 ...

hoping that all of them are prime. Following Euler, we will see in another entry to this blog that this is not so. At this time let's just compare the Euclid numbers e(n) with Fermat numbers f(n). Surprisingly, they are quite similar in more than one way:

• f(0) = 3
• f(n+1) = (f(n) - 2) ⋅ f(n) + 2
• f(n+1) = 2 + ∏_k=0...n f(k)
  

for every n=0 1...

Polya has used the last formula to partially vindicate Fermat's hope—as in the case of the Euclidean sequence, also every two different Fermat numbers are relatively prime (hence once again we see that there are infinitely many different prime numbers). It follows that:

• p(n) ≤ f(n-1)     for every n=1 2...
  

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The iterative formula for  f(n+1)  can be rewritten equivalently as follows:

• f(n+1) = (f(n) - 1)² + 1

for every n=0 1...