I'm taking a discrete math course, and we just started proof writing. I'm struggling a little bit with getting the concepts to click when it comes to proving expressions with exponents.
Here's the homework question I'm stuck on. We are either supposed to put "F" if it can be proven to be false, or give an example of the smallest value for which it's true, if it can be proven true.
∃n∈N(¬Prime(2^(2^n) + 1))
I negated the problem and turned it into this:
∀n∈N(Prime(2^(2^n) + 1))
Then I reasoned something like this:
Given a prime number x, x is the product of two positive integers k , j ∈ N By definition of a prime number, either k = 1 and j = x, or k = x and j = 1.
Because 2^n is an integer (side note: I know this to be true, but I don't know by what theorem or proof, and I think I should list it, whatever it is), let m = 2^n.
Thus, x = (2^m) + 1.
Okay, guys. I know how to prove that this is odd, but I don't know how to prove or disprove that it's prime. The only way I know to disprove a prime is to factor a quadratic.
Can anyone guide me towards where to go from here, and/or correct me if I've made a mistake with what I have so far?
Thank you very much in advance