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Abstract
The polynomial has a unique positive root known as plastic number, which is denoted by
and is approximately equal to 1.32471795. In this note, we study the zeroes of the generalised polynomial
, for
, and prove that its unique positive root
tends to the golden ratio
as
. We also derive bounds on
in terms of Fibonacci numbers.
Keywords:
Public Interest Statement
In this paper, the author considers a result of Siegel (Citation1944), who proved that the positive root of the polynomial , where
is a positive integer greater than or equal to
, tends to the golden ratio
, as
. The positive root of the polynomial
is known as “plastic number” and is the smallest number of the set of algebraic integers whose conjugates are inside the unit circle, known as Pisot–Vijayaraghavan numbers Pisot (Citation1938), Vijayaraghavan (Citation1941).
The submitted proof is elementary and simple. The author also obtains bounds on the positive zero of the polynomial in terms of Fibonacci numbers.
1. Introduction
The recurrence , with initial values
and
yields the celebrated Fibonacci numbers. It is well known that for
where is the positive root of the characteristic polynomial
, known as golden ratio.
One can readily generalise the recurrence and define the order Fibonacci sequence
, with initial conditions
and
. The characteristic polynomial of this recurrence is
. Its zeroes are much studied in literature: we refer to Martin (Citation2004), Miles (Citation1960), Miller (Citation1971), Wolfram (Citation1998), and Zhu and Grossman (Citation2009), where it is proved that the unique positive root tends to
, as
. Series representations for this root are derived in Hare, Prodinger, and Shallit (Citation2014) by Lagrange inversion theorem.
In this note, we turn our attention to the positive zero of the polynomial , known as plastic number, which will throughout be denoted by
and is equal to
(Finch, Citation2003). The plastic number was introduced by van der Laan (Citation1960). The recurrence relation is
, with initial conditions
and defines the integer sequence, known as Padovan sequence Stewart (Citation1996). Although the bibliography regarding the analysis of Fibonacci numbers is quite extensive, it seems not to be this case regarding the plastic number.
In the next section, we examine a generalisation of the Padovan sequence and its associated characteristic polynomial and derive bounds on the unique positive root of the polynomial . The presented study utilises results from the theory of linear recurrences and elementary Calculus, where the nature of roots of this polynomial is investigated. Furthermore, the bounds on the largest root are developed using identities of Fibonacci numbers.
2. The generalised sequence
Consider the recurrence
for and initial conditions
. For
, we obtain as a special case the Padovan sequence. A lemma follows regarding the roots of its characteristic polynomial.
Lemma 2.1
The polynomial has
simple roots. If
is odd, the polynomial has a unique real root
and
complex roots. When
is even, the roots of the polynomial are
,
and
complex zeroes.
Proof
It can be easily seen that neither nor
are roots of
. Following Miles (Citation1960) and Miller (Citation1971), it is convenient to work with the polynomial
(1)
(1)
Differentiating Equation 1, we obtain(2)
(2)
Equation 2 is , at
or at the roots of the quadratic polynomial:
(3)
(3)
Its discriminant can be easily computed to , for all
and the two real roots of polynomial of (3) are
(4)
(4)
We identify the real roots by elementary means. Note that and
and apply Descartes’ rule of signs to Equation 1, there is a unique positive root
in
, and for
even, the unique negative root of the polynomial is
. Also, the polynomial is monic and by Gauss’s lemma the root
is irrational.
Observe that , since
is negative for all
and
. For if they were equal, then by Rolle’s theorem there is at least one root
of Equation 3 in
, but
and considering the fundamental theorem of Algebra, which states that every polynomial with complex coefficients and degree
has
complex roots with multiplicities, we arrive in contradiction. This shows that the polynomial
has
simple roots. We complete the proof noting that
and
are positive and increasing for
and negative for
.
A direct consequence of Lemma 2.1 is
Corollary 2.2
The polynomial is irreducible on the ring of integer numbers
if and only if
is odd.
Further, it is easy to prove that all complex zeroes of the polynomial are inside the unit circle. The next Lemma is from Miles (Citation1960) and Miller (Citation1971).
Lemma 2.3
(Miles, Citation1960; Miller, Citation1971) For all the complex zeroes of the polynomial
, it holds that
.
Proof
Assume that there exists a complex (and hence
), with
. We have that
and
(5)
(5)
Applying the triangle inequality to Equation 5, we deduce that
which contradicts Lemma 2.1. Assuming now that , we have
which is equivalent to and again we arrive in contradiction. Finally, by the same reasoning it can be easily proved that there is no complex zero
, with either
or
.
Lemma 2.3 implies that the solution of the generalised recurrence can be approximated by(6)
(6)
with negligible error term. In Equation 6, is a constant to be determined by the solution of a linear system of the initial conditions.
We now consider, more carefully, Equation 4
Observe that is increasing and bounded sequence. Furthermore,
(7)
(7)
Also, is decreasing and bounded and
(8)
(8)
From Equations 7 and 8, we deduce that two of the critical points of Equation 1, (recall that these are with multiplicity
,
and
), converge to
and
. A straightforward calculation can show that
are points of local minima of the function
to the interval
, so
for all
and by squeeze lemma we have that
.
We remark that is a Pisot–Vijayaraghavan number, a real algebraic integer having modulus greater to
where its conjugates lie inside the unit circle (Bertin, Decomps-Guilloux, Grandet-Hugot, Pathiaux-Delefosse, & Schreiber, Citation1992). These numbers are named after Pisot (Citation1938) and Vijayaraghavan (Citation1941), who independently studied them. Siegel (Citation1944) considered several families of polynomials and showed that the plastic number is the smallest Pisot–Vijayaraghavan number. By Lemmas 2.1 and 2.3, the positive zeroes of the polynomial
, where
is odd, are Pisot–Vijayaraghavan numbers. In case that
is even, the positive roots of the polynomial
are Salem numbers (Salem, Citation1945). This family of numbers is closely related to the set of Pisot–Vijayaraghavan numbers. They are positive algebraic integers with modulus greater than
, where its conjugates have modulus no greater than
and at least one root has modulus equal to
.
We have proved that for all ,
(9)
(9)
Using the identity (Hoggatt, Citation1969, Section 5), we have that for
(10)
(10)
where is the
th Lucas number, defined by
for
, with initial conditions
and
. Lucas numbers obey the following closed form expression for
, (Hoggatt, Citation1969)
Now inequality 10 becomes
Actually, inequality 10 is valid when is quadratic irrational. A stronger result is the following Theorem.
Theorem 2.4
For , it holds that
Proof
For , we have that
where . Since for
,
and
, by Lemma 2.1 it suffices to show that
and
. Setting
to Equation 1, we have to prove that
The previous inequality is the same as(11)
(11)
The left-hand side of inequality 11 is
by Cassini’s identity (Hoggatt, Citation1969) Thus, (11) is true for all .
In order to prove that , we have to equivalently show that
(12)
(12)
We then have
which is
Using that
the identity
can be easily proven by induction, and thus completes the proof.
Acknowledgements
I thank the anonymous referees for their helpful suggestions.
Additional information
Funding
Notes on contributors
Vasileios Iliopoulos
Vasileios Iliopoulos obtained his PhD from the Department of Mathematical Sciences of University of Essex in 2014 and is currently working as a part-time lecturer there. His research interests are on analytic and enumerative combinatorics, operational research and elementary number theory.
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