They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences.
A fundamental property of elliptic divisibility sequences is that they satisfy the general recursion relation (This formula is often applied with r = 1 and W1 = 1.)
Then the sequence grows quadratic exponentially in the sense that there is a positive constant h such that The number h is the canonical height of the point on the elliptic curve associated to the EDS.
He uses the existence of primitive divisors in EDS on elliptic curves of rank one to prove the undecidability of Hilbert's tenth problem over certain rings of integers.
Katherine E. Stange[7] has applied EDS and their higher rank generalizations called elliptic nets to cryptography.
She shows how EDS can be used to compute the value of the Weil and Tate pairings on elliptic curves over finite fields.