This is a problem from the Indian Statistical Institute, ISI BStat 2006 Subjective Problem 3 based on Sophie Germain Identity. Try to solve it.
Prove that is composite for all values of n greater than 1.
Teacher: This problem uses an identity that has a fancy name: Sophie Germain identity. But what’s in a name after all.
Clearly if n is even the expression is composite as it is divisible by 2. We have to check what happens when n is odd.
Student: I remember Sophie Germain’s identity. It says that can be further factorized. As you hinted we can use it here.
Suppose n = 2k +1 (for some k).
Thus we can use identity.