I am making some educated guesses (conjectures) about 3^n-1 and 3^n-2^2 based on making a table in excel and looking for patterns. When are these numbers prime?
Mathematicians look for patterns and then make conjectures which then they try to prove. So, this is what I am doing:
Conjectures about 3^n-1
1) This result is always even and therefore never prime for n > 1
2) After n=32 the result will always end in zero
3) After n=32 the number of zeros will typically increase so that for large numbers of n there will be a lot of zeros. The ratio of numbers other than zero to zero will decrease as n grows.
For n = 66
result = 30903154382632600000000000000000
For n = 200
result = 265613988875875000000000000000000000000000000000000000000000000000000000000000000000000000000000
Conjectures about 3^n-2^n
1) for n > 31 the result will always end in a zero and not be prime
2) The numbers of zeros is increasing with n increasing
Also, I still need to look at the values left here because they appear prime or possibly prime. I need to write a computer program to check that they're prime. However, all other numbers resulting from 3^n-2^n proved to not be prime.