Elliptic Divisibility Sequence - Divisibility Property

Divisibility Property

An EDS is a divisibility sequence in the sense that

m \mid n \Longrightarrow W_m \mid W_n.

In particular, every term in an EDS is divisible by W1, so EDS are frequently normalized to have W1 = 1 by dividing every term by the initial term.

Any three integers b, c, d with d divisible by b lead to a normalized EDS on setting

W_1 = 1,\quad W_2 = b,\quad W_3 = c,\quad W_4 = d.

It is not obvious, but can be proven, that the condition b | d suffices to ensure that every term in the sequence is an integer.

Read more about this topic:  Elliptic Divisibility Sequence

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