1.1. MAIN DEFINITIONS AND PRINCIPAL PROPERTIES 9
continued fraction map and its close relatives may be found in [132], [133], [278],
[279], [552], [638], [639], [901], [1077], [1086], [1151].
More straightforward iterations of the floor function F(X) = |_a-^J
a(x + 1) = [aa(x)\ , xGN ,
are also of interest [402]. The literature contains many further examples: For
instance, recurrence sequences on algebraic varieties [291], [967] and on elliptic
curves [992].
The Lyness sequences arising from the iteration
u(n + 1) + a
and the generalization
u(n + 2) =
u(n + 2)
u(n)
u(n + 1) + a
u(n) -f b
have been extensively studied as dynamical systems and have intimate connections
with elliptic curves (see [766], [648] and [1376] and references for modern treat-
ments.)
1.1.17. Elliptic divisibility sequences and Somos sequences. In the pa-
pers [1329] and [1330], Morgan Ward considered two-sided sequences satisfying the
equation
(1.10) a(x + y)a(x - y) = a(y + l)a(y -
l)a(x)2
- a(x + l)a(x -
l)a(y)2.
Such sequences are called elliptic because they can be parametrized by elliptic
(theta) functions, and there are non-trivial connections between their growth rates
and the canonical height on an associated elliptic curve. There is a surprising
variety of such sequences. The sequences defined by a(x) = x, a(x) = (§) (the
Legendre symbol modulo 3), and
a(x) = ( a ^ ) - * * - 1 ' / 2 ^ ^
a \ OL2
for any ct\ ^0.2, satisfy (1.10). These matters will be expanded on in Chapter 10.
An emerging theory of bilinear recurrence sequences (see Section 1.1.20) may sub-
sume many of the known results about both linear recurrence sequences and elliptic
divisibility sequences.
A Somos-k sequence is one which satisfies the recurrence relation
Lfc/2j
(1.11) a(n)a(n k) = 2 ,
a(n
~
i)a(n
~ k + i), nk.
2 = 1
For example, when k 4, the sequence
(1.12) 1,1,1,1,2,3,7,23,...
occurs frequently as a test sequence for conjectures. See Section 11.1.3 below for
a recent combinatorial interpretation of this sequence. The recurrence (1.11) with
initial values a(0) = = a(k 1) = 1, is an integer sequence for k = 4, 5, 6, 7 but
not for k = 8 [1084], see also [499, Sect. E15]. The thesis of Christine Swart [1268]
proves some of Raphael Robinson's conjectures from [1084]. For example, it is
known that not all primes can divide a term of (1.12) for example 5 and 29
divide no term. In [1084] it is conjectured that if p is an odd prime dividing a term
Previous Page Next Page