p-adic Properties of Translated Division Polynomials and Somos Sequences

In this paper we consider the sequences $({\mathrm{\Psi }}_n(\mathbf{P}){)}_{n\ge 0}$, $({\mathrm{\Phi }}_n(\mathbf{P}){)}_{n\ge 0}$ and $(\bar{\mathrm{\Omega }}{}_n(\mathbf{P}$ of values of the translated division polynomials of an elliptic curve $E\mathrm{/}K$ evaluated at a point $\mathbf{P}\in$ $E(K{)}^2$. We prove that these sequences are purely periodic when $K$ is a finite field. Then we use the periodicity properties of these sequences to prove that certain subsequences of these sequences are $$-Cauchy. Finally, we use this result to prove analogous results for Somos $4$ and Somos $5$ sequences.