A FULL CHARACTERISATION OF DIVISIBILITY SEQUENCES ...

2 Bézivin, Pethő and van der Poorten
We rehearse a number of well known facts, principally to set our notation and terminology,
but also because this is a field in which it is peculiarly difficult to find congenial summaries
that warrant citing.
The present paper arises from discussion at the First Conference of the Canadian Number
Theory Society, Banff 1988, between the latter two authors (see [14],§6.7) and independent
contemporaneous work of the first author [1].
Of course, we are aware that, in the literature, the notion “divisibility sequence” need not
entail that the sequence be a recurrence sequence. Thus some qualifying adjective should
have been attached to each use of the phrase “divisibility sequence”. However, since our
notion “divides” is more general than in the classical case, we feel we can indulge in the
luxury of avoiding a repeated qualifier and may allow our notion “divisibility sequence” to
be more restrictive than needs be.
1. Introduction
1.1 A Divisibility Sequence
Consider the Fibonacci numbers (fh) , defined by the recurrence relation fh+2 = fh+1 + fh
and the initial conditions f0 =0, f1 =1:
h 0123456 7 8 9 1011 12 13 14 15 16 17 18 19 20 ...
fh 011235813213455891442333776109871597 2584 4181 6765 . .....
One notices such phenomena as 13 | 377 and 55 | 6765 — the reason is that, respectively,
7 | 14 and 10 | 20 . Indeed, the Fibonacci sequence has the interesting property that h | k
implies fh | fk .
1.2 Divisibility Sequences
Let ahX h represent a rational function defined over a field F of characteristic zero
and vanishing at ∞ . Then the sequence of Taylor coefficients (ah) is called a divisibility
sequence (or a suite récurrente arithmétique) if the set of quotients {ak/ah : h | k} is a
subset of a ring R finitely generated over Z ,orof finite type over Z . (Should ah = ak =0,
it is useful to define the quotient, say to be 0 .) In the classical, and simpler, formulation
one supposes the ah to be rational integers and for the quotients to belong to Z . However,
the new setting introduces no new problems and allows our proof to deal with various
generalisations of the original question (cf [8],[24]) that have appeared in the literature.
Ward (cf [22]) adds the qualifier linear to emphasise that his divisibility sequences satisfy
a linear homogeneous recurrence relation with constant coefficients. Elsewhere, [24]
he refers to such divisibility sequences as Lucasian sequences; this terminology has not
persisted.
If, moreover, one always has gcd(ah,ak) =agcd(h,k) , perhaps expressed as an equality
of ideals, then the sequence (ah) is called an strong divisibility sequence. Schinzel [20]
characterises all second order such sequences in algebraic number fields.
The reader will note that the divisibility property of itself does not guarantee an especially
interesting sequence. However quite amusing sequences arise from iteration: for if both (ah)
and (bh) are divisibility sequences and the bh all are nonnegative rational integers, then,
plainly, (abh ) has the divisibility property. See also Ward [25].