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 Fn = 22n + 1 is prime for every integer n. These values Fn 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, Fn 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 2m + 1, then m is a power of 2.
(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:
xk + 1 = (x + 1)(xk-1 - xk-2 + ... - x + 1) [See Lemma 4, here where a=x, b=1]
(4) Let x= 2m/k
(5) This then gives us:
(2m/k)k + 1= (2m/k + 1)([2m/k]k-1 - [2m/k]k-2 + ... - [2m/k] + 1)
(6) Since (2m/k)k + 1 = 2m + 1, it follows that 2m + 1 cannot be a prime since it is divisible by (2m/k + 1).
(7) Therefore, we have a contradiction and we reject our assumption in step #1.
- Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001