Infinitude of Primes
Via Fermat Numbers

The Fermat numbers form a sequence in the form F_n = 2²ⁿ + 1, n = 0, 1, 2, ... Clearly all the Fermat numbers are odd. Moreover, as we'll see shortly, any two are mutually prime. In other words, each has a prime factor not shared by any other. Hence, the number of primes cannot be finite.

That no two Fermat numbers have a non-trivial common factor follows from the two lemmas below.

Lemma 1

Proof

The proof is by induction. Since F₀ = 3 and F₁ = 5, (1) is indeed true for k = 1. Assume it holds for k = n. For k = n+1,

Lemma 2

For n ≠ m, F_n and F_m are mutually prime.

Proof

Indeed, assume t divides both F_n and F_m, m < n. By Lemma 1,

F_n = F₀·F₁·...·F_m·...·F_n-1 - 2,

implying that t divides

F₀·F₁·...·F_m·...·F_n-1 - F_n = 2.

But 2 has only two factors: 1 and 2. Being odd, Fermat numbers are not divisible by 2. Hence t = 1, which proves the lemma.

Reference

1. M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 2000
2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford Science Publications, 1996.

• Infinitude of Primes
  1. Infinitude of Primes - A Topological Proof
  2. Infinitude of Primes - A Topological Proof without Topology
  3. Infinitude of Primes Via *-Sets
  4. Infinitude of Primes Via Coprime Pairs
  5. Infinitude of Primes Via Fermat Numbers
  6. Infinitude of Primes Via Harmonic Series
  7. Infinitude of Primes Via Lower Bounds
  8. New Proof of Euclid's Theorem

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