Elliptic Divisibility Sequence - Definition

Definition

A (nondegenerate) elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n ≥ 1 defined recursively by four initial values W1, W2, W3, W4, with W1W2W3 ≠ 0 and with subsequent values determined by the formulas

\begin{align} W_{2n+1}W_1^3 &= W_{n+2}W_n^3 - W_{n+1}^3W_{n-1},\qquad n \ge 2, \\ W_{2n}W_2W_1^2 &= W_{n+2}W_n W_{n-1}^2 - W_n W_{n-2}W_{n+1}^2,\qquad n\ge 3,\\ \end{align}

It can be shown that if W1 divides each of W2, W3, W4 and if further W2 divides W4, then every term Wn in the sequence is an integer.

Read more about this topic:  Elliptic Divisibility Sequence

Famous quotes containing the word definition:

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)