How to find the biggest prime numbers

Along these lines, it has been argued that the ancient Greeks could have discovered Lucas-Lehmer with their contemporary technology: https://infoscience.epfl.ch/record/215073/files/MGFinal.pdf

2^n + 1

not always prime: try n=3 or n=5 or n=6 or n=7 or n=9 or n=10 or ... fun fact: all of these counterexamples have been divisible by 3 or 5.

Brain sending a whole load of 'nope' signals this morning - hope to come back to this later :)