Those who love (elementary) problem-solving eventually come across the Sophie Germain identity. It has lots of applications in elementary number theory, algebra and so on. The identity states
Indeed, note that
Let us use the above identity in solving a couple of problems. Here is the first one.
Problem 1: Evaluate .
Solution: A first glance tells us that the sum should somehow “telescope.” But the denominator looks somewhat nasty! And, it is here that the above identity comes to our rescue. Using the Sophie Germain identity, we first note that
We thus have
Here is another one.
Problem 2: Show that is a prime iff , where .
Solution: Note that if is even, then the expression is clearly composite. If is odd, say, for some , then we have
And, if , then both the factors above are greater than , and hence the expression is composite. Moreover, if , we have , which is prime. And, we are done.