towers of 2 and primes, generation s of big primes by iteration of the function $2^{x}$

Question

let be $2+1=3 $ , $ 2^{2}+1=5 $ , $ 2^{2^{2}}+1= 17 $

apparently what would be the first counterexample ??

is almost true that towers of 2 (iterations of the function $ 2^{x}$

will give big primes ?

Answer 1

Short Answer : Even if this gives only prime numbers, we can't check for more than $5$ iterations, whether the number is prime or not. Our computers can't (at least for now)

Reason : As of January 2018, the largest known prime number is $2^{77,232,917} − 1$, a number with $23,249,425$ digits. It was found by the Great Internet Mersenne Prime Search (GIMPS).

On the other hand, the number, $2^{2^{2^{2^{2^{2}}}}}+1$ (Don't count, $2$ appears six times) has somewhere around $6 \times 10^{19727}$ digits.

Answer 2

$2\uparrow\uparrow4+1 = 65537$ is prime.

However, the next step is a counterexample: $2\uparrow\uparrow5+1$ is divisible by 825753601.