Determining the number of twin primes is one of the most easily-understood

open questions in mathematics, and one of the hardest to answer.

Nevertheless, there are many facts about twin primes that we can prove, and

there are many interesting conjectures to explore. We shall examine several

of these, and then focus on one fairly new method of estimating how many

twin primes there are. We shall show that the sum of the reciprocals of the

twin primes converges, and then provide a new explicit upper bound on this

sum, called Brun's Constant.