Disproving the claim that the numbers 1+2+4, 1+2+4+8, 1+2+4+8+16... alternate between prime and composite I am working through an elementary number theory book and I have come across the following  problem.

Show the following claims are wrong:
Claim 1: The sequence 1+2+4, 1+2+4+8, 1+2+4+8+16, ... is alternately prime and composite.
Claim 2: One or both of the numbers 6n-1 and 6n+1 are prime.

I have actually found counterexamples to both claims, but it was only through cold hard number-crunching. I want to believe this problem wouldn't be here unless there was supposed to be some elegant way to disprove these claims. Is there some other way to disprove the claims other than just finding a counterexample?
Note: the two claims are not meant to be related in any way.

Number crunching is a time-honored technique; many conjectures are far more easily solved by actually testing them (either in a brute-force fashion or by more clever searches) rather than thinking hard about them.
But sometimes these conjectures -- as well as the methods of disproving them -- suggest more general statements. The numbers in your first conjecture are
$$ \sum_{i=0}^{n-1} 2^i = 2^n - 1$$
are called Mersenne numbers. As you've noted, it's easy enough to prove they don't alternate prime/composite, and there are a number of theorems about when a Mersenne number can be prime or composite (e.g. for $2^n - 1$ to be prime, $n$ must also be prime) 
There are also open questions -- e.g. nobody knows whether or not there are infinitely many Mersenne numbers that are prime.
