Determine whether 6211 1 1 is primeSolution211 1 is prime

Determine whether 6^((2^11) - 1)) + 1 is prime.

Solution

(2^11) - 1) is prime since 11 is prime.

Now we can form the number Q by multiplying together all these primes and adding 1, so

Q = (P1 × P2 × P3 × P4... × Pn) + 1

Now we can see that if we divide Q by any of our n primes there is always a remainder of 1, so Q is not divisible by any of the primes.

But we know that all positive integers are either primes or can be decomposed into a product of primes. This means that either Q must be prime or Q must be divisible by primes that are larger than Pn.

hence 6^((2^11) - 1)) + 1 is not prime.