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In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in Galois extensions of prime degree. We also study growth of the Shafarevich-Tate group in cyclic degree p-extensions and improve upon previously known results in this direction.

In this paper, we determine all primitive solutions to the equation (x + r) 2 + (x + 2r) 2 + • • • + (x + dr) 2 = y n for 2 ≤ d ≤ 10 and for 1 ≤ r ≤ 10 4 (except in the case d = 6, where we restrict 1 ≤ r ≤ 5000). We make use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.