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.**

Discussion:

**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).

So

Thus we can use identity.

So

## Some Useful Links:

Solving a few Diophantine Equations – Video

ISI 2015 BStat – BMath Objective Problems

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