Fermat Numbers

While of all Pierre de Fermat's proposed theorems really turned out to be theorems, not all of his conjectures turned out to be true.

In all fairness to Fermat, these theorems and conjectures were published after his death and were based on the notes he kept.

Pierre de Fermat conjectured that the F_n = 2²ⁿ + 1 is prime for every integer n. These values F_n are today called Fermat Numbers.

The content in today's blog is taken from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

Now, for n=0, 1, 2, 3, 4 the results are F₀=3, F₁=5, F₂=17, F₃=257, and F₄=65537 which are all prime.

In 1732, Leonhard Euler showed that F₅ = 641*6,700,417.

Since then, it has been shown that for 5 ≤ n ≤ 16, F_n is not prime.

Despite all these efforts, no other Fermat number has been found to be prime and no proof has been offered that would resolve this issue.

I will end today's blog with the following proof:

Lemma 1: If a prime number p has the form 2^m + 1, then m is a power of 2.

Proof:

(1) Assume that m is not a power of 2.

(2) Then it is divisible by some odd integer k.

(3) Now, we know that:

x^k + 1 = (x + 1)(x^k-1 - x^k-2 + ... - x + 1) [See Lemma 4, here where a=x, b=1]

(4) Let x= 2^m/k

(5) This then gives us:

(2^m/k)^k + 1= (2^m/k + 1)([2^m/k]^k-1 - [2^m/k]^k-2 + ... - [2^m/k] + 1)

(6) Since (2^m/k)^k + 1 = 2^m + 1, it follows that 2^m + 1 cannot be a prime since it is divisible by (2^m/k + 1).

(7) Therefore, we have a contradiction and we reject our assumption in step #1.

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001