Fermat Numbers

Fermat Numbers, a class of numbers, are the integers of the form $ F_n=2^{2^n} +1 \ \ n \ge 0$ .

For example: Putting $ n := 0,1,2 \ldots$ in $ F_n=2^{2^n}$ we get $ F_0=3$ , $ F_1=5$ , $ F_2=17$ , $ F_3=257$ etc.

Fermat observed that all the integers $ F_0, F_1, F_2, F_3, \ldots$ were prime numbers and announced that $ F_n$ is a prime for each natural value of $ n$ .

In writing to Prof. Mersenne, Fermat confidently announced:

I have found that numbers of the form $ 2^{2^n}+1$ are always prime numbers and have long since signified to analysts the truth of this theorem.

However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that $ F_1 -F_4$ are primes but $ F_5=2^{2^5} =4294967297$ is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that $ F_5$ is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.

Theorem: 
The Fermat number $ F_5$ is divisible by $ 641$ i.e., $ 641|F_5$ .
Proof:As defined $ F_5 :=2^{2^5}+1=2^{32}+1 \ \ldots (1)$Factorising $ 641$ in such a way that $ 641=640+1 =5 \times 128+1 \\ =5 \times 2^7 +1$
Assuming $ a=5 \bigwedge b=2^7$ we have $ ab+1=641$ .Subtracting $ a^4=5^4=625$ from 641, we get $ ab+1-a^4=641-625=16=2^4 \ \ldots (2)$ .Now again, equation (1) could be written as
$ F_5=2^{32}+1 \\ \ =2^4 \times {(2^7)}^4+1 \\ \ =2^4 b^4 +1 \\ \ =(1+ab-a^4)b^4 +1 \\ \ =(1+ab)[a^4+(1-ab)(1+a^2b^2)] \\ \ =641 \times \mathrm{an \, Integer}$
Which gives that $ 641|F_n$ .

Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond $ F_4$ . The best guess is that all Fermat numbers $ F_n>F_4$ are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers $ F_n, F_m \ m > n \ge 0$ , $ \mathrm{gcd}(F_m, F_n) =1$ .

Following two theorems are very useful in determining the primality of Fermat numbers:

Pepin Test:
For $ n \ge 1$ , the Fermat number $ F_n$ is prime $ \iff 3^{(F_n-1)/2} \equiv -1 \pmod {F_n}$
Euler- Lucas Theorem
Any prime divisor $ p$ of $ F_n$ , where $ n \ge 2$ , is of form $ p=k \cdot 2^{n+2}+1$ .

Fermat numbers ($ F_n$ ) with $ n=0, 1, 2, 3, 4$ are prime; with $ n=5,6,7,8,9,10,11$ have completely been factored; with $ n=12, 13, 15, 16, 18, 19, 25, 27, 30$ have two or more prime factors known; with $ n=17, 21, 23, 26, 28, 29, 31, 32$ have only one prime factor known; with $ n=14,20,22,24$ have no factors known but proved composites. $ F_{33}$ has not yet been proved either prime or composite.

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