An elliptic curve E defined over \Q is an algebraic variety which forms a
finitely generated abelian group, and the structure theorem then implies that E
= \Z^r + \Z_{tors} for some r \geq 0; this value r is called the rank of E. It
is a classical problem in the study of elliptic curves to classify curves by
their rank. In this paper, the author uses the method of 2-descent to calculate
the rank of two families of elliptic curves, where E is given by E: y^2 =
x(x-p)(x-2) with p, p-2 being twin primes.