Elliptic divisibility sequence

In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward in the 1940s. 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. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography. 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 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. An EDS is a divisibility sequence in the sense that 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 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. 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.) The discriminant of a normalized EDS is the quantity An EDS is nonsingular if its discriminant is nonzero. A simple example of an EDS is the sequence of natural numbers 1, 2, 3,... . Another interesting example is 1, 3, 8, 21, 55, 144, 377, 987,... consisting of every other term in the Fibonacci sequence, starting with the second term. However, both of these sequences satisfy a linear recurrence and both are singular EDS.

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Publications associées (2)

Prime Power Terms In Elliptic Divisibility Sequences

Valéry Aurélien Mahé

We study a problem on specializations of multiples of rational points on elliptic curves analogous to the Mersenne problem. We solve this problem when descent via isogeny is possible by giving explicit bounds on the indices of prime power terms in elliptic divisibility sequences associated to points in the image of a nontrivial isogeny. We also discuss the uniformity of these bounds assuming the Hall-Lang conjecture.

American Mathematical Society2014

Algebraic Divisibility Sequences Over Function Fields

Valéry Aurélien Mahé

In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

Australian Mathematical Society2012