Lethbridge Number Theory and Combinatorics Seminar: Jeff Bleaney

In 1948, Morgan Ward introduced the concept of an Elliptic Divisibility Sequence (EDS) as an integer sequence $(W_{n})$ which satisfies the recurrence relation $$W_{m+n}W_{m-n}W_{1}^{2} = W_{m+1}W_{m-1}W_{n}^{2} - W_{n+1}W_{n-1}W_{m}^{2},$$ and satisfies the additional property that $W_{m}|W_{n}$ whenever $m|n$. Of particular interest to Ward, were what he called symmetries of an EDS. Ward showed that if $(W_{n})$ is an EDS with $W_{r} = 0$, then we have $$W_{r+i} = ab^{i}W_{i},$$ for some $a$ and $b$.

 

In her Ph.\,D.\ thesis in 2008, Kate Stange generalized the concept of an EDS to an $n$-dimensional array called an Elliptic Net.

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 We will discuss the connections between EDS's, Elliptic Nets, and elliptic curves, and give a generalization of Ward's symmetry theorem for elliptic nets.