1. Consider Pell's equation 22 – Ny2 = 1. Given two solutions (a,b) and (c,d), define their composition as (a,b) * (c,d)

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1 Consider Pell S Equation 22 Ny2 1 Given Two Solutions A B And C D Define Their Composition As A B C D 1 (79.08 KiB) Viewed 50 times

1. Consider Pell's equation 22 – Ny2 = 1. Given two solutions (a,b) and (c,d), define their composition as (a,b) * (c,d) = (ac + Nbd, ad + bc). = Brahmagupta's identity then implies that the composition is also a solution to the equation. (i) Note that (1,0) is a solution. Show that (a,b) * (1,0) = (1,0) * (a,b) = (a,b) = for any solution (2, y) = (a,b) of the equation. (ii) Show that (a,b) * (a, —b) = (a, -b) * (a,b) = (1,0). In modern terminology, this tells us that the solutions to Pell's equation for a fixed N form a group under the operation of composition defined above. Here (1,0) is the identity element of the group and (a, b) the inverse of (a,b).