On the analytic rank of the twin prime elliptic curve y²=x (x-2) (x-p)

Let p 7 and suppose (p, p-2) are twin prime numbers, in Hatley, 2009, the elliptic curve Eₚ: y²=x (x-2) (x-p) was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of Eₚ/Q. I show that for p 3, 5 8, the analytic rank of Eₚ is at least one (Theorem 1. 1. 2) in line with Beers' predictions. This is done by finding a formula (Theorem 4. 1. 1) for the global root number of Eₚ for all twin prime pairs. I also show that Beers' conjecture, that for p 1 8 the rank of Eₚ is two, is false as stated because E₇₃ has rank zero. In the light of Theorem 4. 1. 1, Beers' conjecture needs to be modified: if p 1 8 then the rank of Eₚ is zero or two (Conjecture 5. 3. 1).