# Signatures of discrete time-crystallinity in transport through an open
Fermionic chain

Subhajit Sarkar<EMAIL_ADDRESS>Department of Chemistry, Ben-Gurion
University of the Negev, Beer Sheva, 84105, Israel  School of Electrical and
Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva, 84105,
Israel  Yonatan Dubi<EMAIL_ADDRESS>Department of Chemistry, Ben-Gurion
University of the Negev, Beer Sheva, 84105, Israel  Ilse Katz Center for
Nanoscale Science and Technology, Ben-Gurion University of the Negev, Beer
Sheva, 84105, Israel

###### Abstract

Abstract

Discrete time-crystals are periodically driven quantum many-body systems with
broken discrete-time translational symmetry, a non-equilibrium steady state
representing self-organization of motion of quantum particles. Observations of
discrete time-crystalline order are currently limited to magneto-optical
experiments. Crucially, it was never observed in a transport experiment
performed on systems connected to external electrodes. Here we demonstrate
that both discrete time-crystal and quasi-crystal survive a very general class
of environment corresponding to single-particle gain and loss through system-
electrode coupling over experimentally relevant timescales. Using dynamical
symmetries, we analytically identify the conditions for observing time-
crystalline behavior in a periodically driven open Fermi-Hubbard chain
attached to electrodes. Remarkably, the spin-polarized transport current
directly manifests the existence of a time-crystalline behavior. Our findings
are verifiable in present-day experiments with quantum-dot arrays and
Fermionic ultra-cold atoms in optical lattices.

## Introduction

Spontaneous symmetry breaking represents a unifying concept which ubiquitously
spans from condensed matter and atomic physics to high energy particle physics
Anderson (1972). Examples include, superconductors, Bose-Einstein condensates,
(anti)ferromagnets, all the crystals, and even (Higgs) mass generation for
fundamental particles Bardeen  _et al._ (1957); Anderson  _et al._ (1995);
Davis  _et al._ (1995); Chaikin and Lubensky (1995); Cottingham and Greenwood
(2007). However, time-translation symmetry has always been special:
Schödinger’s equation, which governs quantum physics, is indeed time-
translation invariant. In spite of that, time-translation symmetry breaking
has been shown to be possible under special circumstances, leading to the
discrete time-crystal (DTC) behavior. Periodically driven (Floquet) closed
quantum systems that never reach a thermodynamic equilibrium can indeed
exhibit a DTC behavior Sacha and Zakrzewski (2017); Khemani  _et al._ (2016);
Else  _et al._ (2016); Surace  _et al._ (2019); Khemani  _et al._ (2019);
Wilczek (2012); Choi  _et al._ (2017); Zhang  _et al._ (2017). The breaking of
discrete time translation symmetry is manifested in the sub-harmonic
oscillations of the order parameter and typically rely on disorder and
localization to avoid reaching a stationary state of infinite temperature that
opposes time crystalline order Yao and Nayak (2018); Bukov  _et al._ (2015);
Khemani  _et al._ (2016); Moessner and Sondhi (2017). The first signature of
DTC was reported in strongly interacting spin systems Choi  _et al._ (2017);
Zhang  _et al._ (2017), measured using spin-dependent fluorescence (optical
measurements) Choi  _et al._ (2017); Zhang  _et al._ (2017); Kyprianidis  _et
al._ (2021). Subsequently, DTCs have been realized in variety of systems Dogra
_et al._ (2019); Pal  _et al._ (2018); Träger  _et al._ (2021); Smits  _et
al._ (2018); Autti  _et al._ (2021). Recent advances in the fabrication and
control of exchange coupled quantum-dot arrays Ito  _et al._ (2018);
Mukhopadhyay  _et al._ (2018); Mills  _et al._ (2019); Sigillito  _et al._
(2019); Qiao  _et al._ (2020) led to the theoretical proposal of observing a
DTC Barnes  _et al._ (2019) and subsequent experimental realization of the
same leading to stable quantum information processing Qiao  _et al._ (2021);
Van Dyke  _et al._ (2021); Estarellas  _et al._ (2020).

Observations of DTCs has been limited to closed Floquet quantum systems (i.e.,
periodically driven quantum system without dissipation), as a transient
phenomenon restricted to optical detection from spin-dependent fluorescence or
magnetization measurements Choi  _et al._ (2017); Zhang  _et al._ (2017);
Kyprianidis  _et al._ (2021); Mi  _et al._ (2021). Consequently, in spite of
the recent observation of DTC in various platforms Choi  _et al._ (2017);
Zhang  _et al._ (2017); Kyprianidis  _et al._ (2021); Pal  _et al._ (2018);
Träger  _et al._ (2021); Smits  _et al._ (2018); Qiao  _et al._ (2021); Van
Dyke  _et al._ (2021) the experimental observations of DTCs in dissipative
quantum many body systems rarely has been reported until recent observation in
the atom-cavity system Keßler  _et al._ (2021).

Opening the system to an environment indeed presents a challenge of defining
the notion of DTC. Nevertheless, plausible criteria, based on the spectrum of
the Floquet map (i.e., the one period time evolution operator), to define and
characterize DTC, has been put forward Riera-Campeny  _et al._ (2020); Gong
_et al._ (2018) in open systems governed by a Markovian Lindblad master
equation Lindblad (1976); Gorini  _et al._ (1976); Breuer and Petruccione
(2004); Purkayastha  _et al._ (2016). It relies on the existence of several
non-decaying states of the Floquet map with eigen-values
$\mathcal{E}_{DTC}\in\\{\mathcal{E}_{\mu}\\}$ such that $\mathcal{E}_{DTC}\neq
1$ but $(\mathcal{E}_{DTC})^{p}=1$ for some integer ‘$p$’ Riera-Campeny  _et
al._ (2020). Coupling to the external environment makes the DTCs fragile
Lazarides and Moessner (2017); Choi  _et al._ (2017); Zhang  _et al._ (2017),
but dissipation engineering has shown the promise. In this regard, the
mechanisms that stabilize DTCs in open quantum systems can be classified into
three broad categories. These are, mean-field based DTCs which are stable only
in the thermodynamic limit Iemini  _et al._ (2018); Carollo and Lesanovsky
(2022); Gong  _et al._ (2018); Zhu  _et al._ (2019); symmetry-based mechanism
Buča  _et al._ (2019); Chinzei and Ikeda (2020); and meta-stable DTCs that
neither require any symmetry nor disorder Gambetta  _et al._ (2019).

The mean-field-based DTCs require a well-defined semi-classical limit and are
found to be fragile to quantum fluctuations Gong  _et al._ (2018); Zhu  _et
al._ (2019); Keßler  _et al._ (2020); Riera-Campeny  _et al._ (2020). The
symmetry-based mechanism can accommodate quantum fluctuations. However, it
requires a very specific choice of system-environment coupling that satisfies
the existence of dark state space to stabilize DTCs in a dissipative quantum-
many body system Chinzei and Ikeda (2020); Buča  _et al._ (2019); Medenjak
_et al._ (2020a); Buča and Jaksch (2019); Medenjak  _et al._ (2020b); Tindall
_et al._ (2020). Indeed, as detailed in Ref. Buča  _et al._ , 2019,
dissipation (in the form of on-site dephasing) is essential for stabilizing
the DTC behavior, because it allows for coupling (and hence synchronization)
of different sectors in the Floquet Hamiltonian which would otherwise remain
decoupled in absence of dephasing.

However, most of the realistic environment - specifically, a transport setup
in which the system is open to charge transfer from electrodes - does not
satisfy the dark state criterion. How DTCs survive or die out due to coupling
with this type of environment is currently unknown. We show that a DTC and
discrete-time quasi-crystal (DTQC) Pizzi  _et al._ (2019); Giergiel  _et al._
(2019); Zhao  _et al._ (2019) can indeed survive a very general class of
environment, viz., the single-particle transport through system-electrode
coupling, over a sufficiently long time when the system environment coupling
is weak. With stronger system-lead coupling, the system reaches a set of
transient states distinct from the usual Floquet steady-state (FSS).
Strikingly enough, the spin-polarized transport current through the
paradigmatic Floquet Fermi-Hubbard chain connected to electrodes can manifest
DTC/DTQC order, linking the DTC physics with electrical transport.

In the following, we first establish, supported by numerical calculations,
that even if the system is connected to external electrodes (leads), there
emerges a unique “weak local symmetry,” to be described later, that remains
preserved as long as the system-lead coupling is weak. The DTC (and the DTQC)
manifests itself in a sharp peak of the Fourier transform of the expectation
value of the spin-current that respects the “weak local dynamical symmetry”.
The temporal oscillation of the spin-current is locked at a frequency which
provide a direct measure to the DTC (and the DTQC) sub-harmonic frequency. For
strong system-lead coupling both DTC and DTQC indeed decay, however, for an
intermediate strength the system reaches a (possibly degenerate) transient
manifold Macieszczak  _et al._ (2016) and show long lived (and arguably pre-
thermal Kyprianidis  _et al._ (2021); Mi  _et al._ (2021)) DTC(DTQC) behavior
before it decays. To this end we clarify that the notion of transient manifold
we use pertains to the finite lifetime of the DTC, in contrast to the meta-
stable manifold corresponding to the vanishing gap of the spectrum of
Liouvillian/ Floquet map Gambetta  _et al._ (2019). We further show, from the
spectrum of the Floquet map, that the decay of DTC (DTQC) scales linearly with
the system-electrode coupling with the slope being twice the driving period.
The phase of DTC eigenvalues of the Floquet map further shows the frequency
locking phenomena observed in the oscillating spin-current. This frequency
locking is a consequence of the emergence of the weak local dynamical
symmetry. The scaling behavior and frequency locking phenomena we obtain is
independent of the values of the system parameters, the magnitude of the
external drive, the choice of the initial state, and system size.

## Results and Discussions

### System

In order to be specific, we consider a quantum-dot array consisting of $N$
dots (numerical simulations are done with $N=3$) attached to external
electrodes, see Fig. 1. Such a quantum-dot array set-up can realize a one-
dimensional Fermi-Hubbard model Hensgens  _et al._ (2017). However, our
results are equally valid for Fermi-Hubbard model routinely realized in
optical lattices Gross and Bloch (2017). Each site of the Fermi-Hubbard chain
is irradiated with a laser (electromagnetic wave) of frequency $\omega$ whose
magnetic component of field strength $B$ affects only the spin-dynamics of the
system. Apart from its connection to the electrodes we also subject the system
to onsite dephasing. As we shall show, the magnitude of the onsite dephasing
has no effect of the DTC(DTQC) sub-harmonic frequency. The Hamiltonian of the
system is described in the methods section. We start by analyzing the
Markovian dynamics of the open Fermi-Hubbard chain within the Lindblad master
equation Lindblad (1976); Gorini  _et al._ (1976); Breuer and Petruccione
(2004); Purkayastha  _et al._ (2016), see methods for further details.

Figure 1:  Schematic arrangement of the quantum-dot array set-up.
$t,U,~{}\text{and}~{}K$ are the hopping, onsite and nearest-neighbor
interactions, respectively. $V(t)$ represents the external time-periodic
drive. $\gamma_{L}$ and $\gamma_{R}$ are the system left-lead and system
right-lead couplings, respectively (note the direction of the arrow indicating
the one-way electron transfer, i.e., an infinite bias condition) which is
taken to be the same for both spin-up and spin-down electrons. Onsite
dephasing is given by $\Gamma$.

### Weak local Floquet-dynamical symmetry

In absence of system-lead coupling (i.e., $\gamma_{L}=\gamma_{R}=0$) the
appearance of the TC in the system is stabilized by the presence of a Floquet-
dynamical symmetry (FDS) Neufeld  _et al._ (2019); Chinzei and Ikeda (2020);
Buča  _et al._ (2019); Medenjak  _et al._ (2020a); Buča and Jaksch (2019);
Medenjak  _et al._ (2020b); Tindall  _et al._ (2020), defined as follows.
Given a Floquet map
$\hat{\mathcal{U}}_{F}=\mathcal{T}\left(\text{exp}\left[\int_{0}^{T}\hat{\mathcal{L}}_{s}ds\right]\right)$
[and
$\hat{\mathcal{U}}_{F}^{-1}=\mathcal{T}\left(\text{exp}\left[-\int_{0}^{T}\hat{\mathcal{L}}_{s}ds\right]\right)$
where $\mathcal{T}$ is time ordering] if there exists an operator $A$ (in this
case it is the total spin raising operator $S^{+}=\sum_{j}S_{j}^{+}$) which
satisfies
$\hat{A}(T)=\hat{\mathcal{U}}_{F}\hat{A}\hat{\mathcal{U}}_{F}^{-1}=e^{i\lambda
T}\hat{A}$, and all the Lindblad operators $V_{\mu}$ satisfy
$[V_{\mu},A(t)]=[V_{\mu}^{\dagger},A(t)]=0$, then the system exhibits the FDS
and $A(t)$ oscillates in the long-time limit. We follow the notations:
$\hat{A}$ represents a super-operator which operates on the vectorized density
matrix $|\rho\rangle\rangle$, and $A$ is a normal operator which acts on the
desnity matrix as $A\rho$ in the form of a matrix multiplication, where the
density matrix $\rho$ satisfies Lindblad equation, (5). While the oscillations
in $A(t)$ were shown to be protected against dephasing Chinzei and Ikeda
(2020), this is not the case for system-lead coupling; this form of
dissipation breaks the FDS.

Even if the FDS is not fully protected against the system-lead coupling, a DTC
behavior can still emerge as a transient phenomenon. To demonstrate this,
consider the “local” operator $S_{loc}^{+}=\left(S_{1}^{+}+S_{N}^{+}\right)$,
which satisfies

$(\mathcal{U}_{F})^{p}(S_{loc}^{+}\rho)=e^{ip\lambda
T(1+i\gamma/\lambda)}S_{loc}^{+}(\mathcal{U}_{F})^{p}(\rho),$ (1)

[see Supplementary Note 1 for a proof] after $p$ driving periods for density
matrix $\rho$ satisfying Lindblad equation (5), given
$[V_{L(R)},S_{loc}^{+}]\propto\gamma$ ($\gamma_{L}=\gamma_{R}=\gamma$ is
assumed). In the long time limit (corresponding to $p\rightarrow\infty$) (1)
represents an approximate FDS as long as $\gamma\ll\lambda$. We call it a
“weak local Floquet dynamical symmetry” (weak local FDS) an emergent FDS in
the long time limit near FSS (see Supplementary Note 2). The notion of the
locality comes from the fact that
$S_{loc}^{+}=\left(S_{1}^{+}+S_{N}^{+}\right)=([S_{1}^{+}\otimes
I_{2}\otimes\cdot\cdot\cdot\otimes I_{N}]+[I_{1}\otimes
I_{2}\otimes\cdot\cdot\cdot\otimes S_{N}^{+}])$ is local because this is
diagonal in the site-basis. The notion of weak dynamical symmetry comes from
the fact that $[V_{L(R)},S_{loc}^{+}](\propto\gamma)\neq 0$. The local
(unitary) operator $S_{loc}^{+}$ satisfies
$\mathcal{U}_{F}([S_{loc}^{+}]^{m}\rho[S_{loc}^{+\dagger}]^{n})=e^{i(m-n)\lambda
T}e^{-(m+n)\gamma
T}[S_{loc}^{+}]^{m}\mathcal{U}_{F}(\rho)[S_{loc}^{+\dagger}]^{n}$, (see
Supplementary Note 1 for a proof) a Floquet analogue of the weak dynamical
condition put forward in Ref. Buča and Prosen, 2012, see Supplementary Note 3.
For $\gamma\approx\lambda$ the FDS is indeed not preserved during the time
evolution in the sense that any oscillation in the observable would decay in
their respective amplitudes. However, the frequency of the oscillation remains
unchanged even at $\gamma\approx\omega$ (see Fig. 3).

In the Floquet basis (i.e., in the rotating frame) the equation of motion for
$S^{+}_{loc}$ can be shown to be generated by a Floquet Lindbladian
$\mathcal{L}_{F}$ whose coherent part is governed by a Floquet Hamiltonian
$\mathcal{H}_{F}=\mathcal{H}_{0}+\mathbf{h}\cdot\mathbf{S}$, where
$\mathbf{h}\cdot\mathbf{S}$ is a Zeeman term arising from an effective
homogeneous and static magnetic field, $\mathbf{h}=(B,0,\omega)$ at each site
[see Supplementary Note equations S10]. Given such a Zeeman term we construct,
from the local operator
$S_{loc,|\mathbf{h}|}^{+}=(S_{1,|\mathbf{h}|}^{+}+S_{N,|\mathbf{h}|}^{+})$ in
the Floquet basis (rotating the axis of quantization along $\mathbf{h}$), a
set of coherent states
$\rho_{m,n}=(S_{loc,|\mathbf{h}|}^{+})^{m}\rho_{\text{FSS}}(S_{loc,|\mathbf{h}|}^{-})^{n}$
(with integer values of $m,n$) from the Floquet steady state
$\rho_{\text{FSS}}$, satisfying,

$\mathcal{U}_{F}(\rho_{m,n})=e^{i(m-n)\lambda T}e^{-(m+n)\gamma T}\rho_{m,n},$
(2)

where $\lambda=|\mathbf{h}|=\sqrt{B^{2}+\omega^{2}}$ (modulo $\omega$), and
$\rho_{m,n}$ are the Floquet coherent states with an oscillatory component
(with a characteristic energy scale $\lambda$) and a decaying part (with a
characteristic energy scale $\gamma$). Here $\lambda$ satisfies the usual FDS
structure Chinzei and Ikeda (2020), viz., if $\lambda/\omega=q/p$ with co-
prime integers $p\geq 2$ and $q$, a DTC emerges with a period $T_{DTC}=pT$,
otherwise DQTC emerges, and $\gamma/\lambda$ determines the decay of the of
the DTC (and DTQC).

For $\gamma\neq 0$, (2) shows that the Floquet steady state (FSS) corresponds
to $m=n=0$, i.e., $\rho_{0,0}=\rho_{\text{FSS}}$, and the rest of the states
corresponding to $m=n$ are purely decaying states (in the sense that these do
not exhibit any coherent part). However, for $\gamma=0$ there exists multiple
FSS corresponding to $m=n$ Chinzei and Ikeda (2020). Clearly for $m\neq n$ and
$\gamma=0$, (2) indicates $\rho_{m,n}$ are degenerate eigenstates of
$\mathcal{U}_{F}$ since the same value of $(m-n)$ can be obtained from
different combinations of $m,n$-pair. Finite value of $\gamma$ lifts this
degeneracy.

A DTC density matrix (as well as DQTC) can be written as
$\rho_{DTC}=\rho_{FSS}+\sum_{\begin{subarray}{c}m,n\\\ m,n\neq
0\end{subarray}}c_{m,n}\rho_{m,n}$, where
$c_{m,n}=|\text{Tr}[\rho_{\text{in}}^{\dagger}\rho_{m,n}]|$,
$\rho_{\text{in}}$ being the density matrix of the initial state, are the real
coefficients of the superposition. For $\gamma\ll\lambda$, only a few coherent
states $\rho_{m,n}$ decay sufficiently slowly and most of the other coherent
states (corresponding to larger integer values of $n$) decay in the long time
limit. The time evolution of $\rho_{DTC}$ is obtained by
$\mathcal{U}(t)\rho_{DTC}[\mathcal{U}(t)]^{-1}$. In the long time limit, the
time dependent dissipative DTC density matrix is given by,
$\rho_{DTC}(t)=\rho_{FSS}+\sum_{\begin{subarray}{c}m,n\\\ m,n\neq
0\end{subarray}}\left(c_{m,n}e^{i(m-n)\lambda
t}\rho_{m,n}+h.c\right)e^{-(m+n)\gamma t}$. Then, observable value of any
operator $S^{\alpha}=\prod_{j}S^{\alpha}_{j}$, corresponding to local operator
$S^{\alpha}_{j}$ acting on $j^{\prime}$th site, follows from $\rho_{DTC}(t)$
as $\langle
S^{\alpha}(t)\rangle=\text{Tr}[S^{\alpha}\rho_{FSS}]+\text{Tr}[S^{\alpha}\rho_{DTC}(t)]$
$=\sum_{\begin{subarray}{c}m,n\\\ m,n\neq
0\end{subarray}}2\text{Tr}[S^{\alpha}\rho_{m,n}]e^{-(m+n)\gamma
t}\cos([m-n]\lambda t)+\text{const.}$. For $\gamma=0$ [no system-lead
coupling], $\langle S^{\alpha}(t)\rangle$ exhibits persistent oscillation, and
for non zero $\gamma$ it develops a decaying envelop with decay time
$\sim\gamma^{-1}$ leading to meta-stable DTC, see Supplementary Note 1.

Figure 2: Spectrum of the Floquet map. (a) All the eigenvalues of the Floquet
map $\mathcal{U}_{F}$, the red dots near the unit-circle in the complex plane
(not on the real axis) are the discrete time crystal (DTC) eigenvalues. (b)
Absolute (Abs) values $\text{Abs}[\mathcal{E}_{DTC}]$ and the imaginary (IM)
part $\text{Im}\left[\mathcal{E}_{DTC}\right]$ of the DTC eigenvalues as a
function of dimensionless system-lead coupling $\gamma/\omega$. Blue
horizontal line represents the peripheral spectrum for which
$|\mathcal{E}_{\alpha}|=1$, the orange horizontal line represents
$\text{Im}\left[\mathcal{E}_{DTC}\right]$ for $\gamma=0$. (c) Deviation
$\Delta\mathcal{E}_{DTC}$ of the DTC eigen-value $\mathcal{E}_{DTC}$ from the
eigenvalue of the Floquet steady state, $\mathcal{E}_{\text{FSS}}$ as a
function of $\gamma/\omega$ on a log-log scale. Slope of the straight-line is
0.15. The straight-line corresponds to the function
$\Delta\mathcal{E}_{DTC}=\left(1-e^{-2\gamma T}\right)\approx 2\gamma T$,
where $T=0.1~{}\mu s$ [see Eqs. S(18) - S(22)].

### Numerical analysis

Following the standard prescription Riera-Campeny  _et al._ (2020), we
numerically evaluate the eigen-spectrum of the Floquet map (taking 3 QDs),
viz., $\mathcal{U}_{F}(\rho)=\mathcal{E}_{\alpha}\rho$, $\alpha$ indexing the
spectrum and plot $\mathcal{E}_{\alpha}$ on the complex plane in Fig. 2(a).
This eigen-value equation defines the DTC (DQTC) eigen-values as
$(\mathcal{E}_{DTC(DTQC)})^{p}=e^{ip\lambda T}=1$, DTC for integer $p>2$ and
DQTC for fractional $p(>2)$, leading the the periodicity of the TC to be
$T_{DTC(DQTC)}=pT$, and the FSS eigen-value as $\mathcal{E}_{\text{FSS}}=1$.
Fig. 2(a) further reveals that the meta-stable Floquet coherent states (red
dots near the unit circle) are distinct from the FSS (red square on the unit
circle) because they never coalesce to the FSS as the system-lead coupling
strength increases, indicating the formation of a distinct transient manifold
Macieszczak  _et al._ (2016) within the full spectrum of $\mathcal{U}_{F}$.

The system parameters chosen for the above numerical calculations are well
within the reach of present day experiments Barthelemy and Vandersypen (2013);
Kouwenhoven  _et al._ (2001); Hensgens  _et al._ (2017); Mills  _et al._
(2019); Zajac  _et al._ (2018). We choose
$t_{hop}=U=K=20\pi~{}\text{MHz}=0.26~{}\mu eV$, all being tunable
electostatically, and $\Gamma=0.1t_{hop}$. The frequency of the external drive
is $\omega=20\pi$ MHz (therefore, a period of $0.1~{}\mu s$) assuming the
typical resolution of nano-seconds in the time-dependent measurements in
experiments with quantum-dot array, and magnitude of the external magnetic
field $B$ is of the order of millitesla.

Corresponding to the specific case of DTC, Fig. 2 (b) shows the absolute value
($\text{Abs}[\mathcal{E}_{DTC}]$) and the imaginary part
($\text{Im}[\mathcal{E}_{DTC}]$) of the DTC eigenvalue $\mathcal{U}_{F}$ as a
function of $\gamma$. For sufficiently small values of $\gamma$,
$\text{Abs}[\mathcal{E}_{DTC}]=1$ showing the DTC eigen-value lies on the
unit-circle of the spectrum, and
$\text{Im}[\mathcal{E}_{DTC}]=\sin\left(\frac{2\pi l}{3}\right)$ shows that
the DTC state has a period $T_{DTC}=3T$. Stronger system-lead coupling leads
DTC eigen-value to move away from the periphery and also introduces a detuning
in the DTC time-period as shown in Fig. 2 (b). Therefore, irrespective of the
choice of the initial state the spectrum supports the DTC with a period $3T$
as long as $\gamma$ remain small enough, and Fig. 2 (b) corroborates (1).

Fig. 2 (c) plots the deviation form the periphery,
$\Delta\mathcal{E}_{DTC}=\left(|\mathcal{E}_{\text{FSS}}|-\text{Abs}[\mathcal{E}_{DTC}]\right)$
as function of $\gamma$, showing how the DTC moves away from the peripheral
spectrum. Since the least decaying coherent states contribute the most to the
DTC, we can therefore identify the DTC density matrix to be
$\rho_{DTC}=\rho_{FSS}+\frac{1}{2}\left(\rho_{1,0}+\rho_{2,1}+\rho_{0,1}+\rho_{1,2}\right)$,
which is further supported by the numerical calculations shown in Fig. 2 (c)
[see supplementary equation S(18) - S(22) for details]. Similarly,
$\rho_{DTC}=\rho_{FSS}+\frac{1}{\sqrt{2}}\left(\rho_{2,0}+\rho_{0,2}\right)$
is also another slowly decaying DTC density matrix that matches the scaling
obtained in Fig. 2 (c), [see discussion after supplementary equation S(22)].
However, to which $\rho_{DTC}$ the system will reach depends on the choice of
the initial density matrix Tindall  _et al._ (2020); Riera-Campeny  _et al._
(2020); Gong  _et al._ (2018).

Figure 3: Phase of the discrete time crystal (DTC) eigenvalues relative to
$\frac{2\pi}{3}$:
$\text{Abs}[\theta_{\mathcal{E}_{\alpha}}(\gamma)-\frac{2\pi}{3}]$, the
absolute (Abs) value of the deviation of the phase
$\theta_{\mathcal{E}_{\alpha}}(\gamma)=\arctan\left({\frac{\text{Im}[\mathcal{E}_{\alpha}]}{\text{Re}[\mathcal{E}_{\alpha}]}}\right)$
of DTC eigenvalues $\mathcal{E}_{\alpha}$ from its value
$\theta_{\mathcal{E}_{\alpha}}(\gamma=0)=\frac{2\pi}{3}$ at $\gamma=0$ as a
function of $\gamma/\omega$ on a $\log-\log$ scale.

An important consequence of the weak-local FDS is the fact that the DTC
oscillation frequency is locked during the time evolution even in presence of
system-lead coupling. To highlight this, in Fig. 3 we plot the phase of the
DTC (complex) eigenvalues,
$\theta_{\mathcal{E}_{\alpha}}(\gamma)=\arctan\left({\frac{\text{Im}[\mathcal{E}_{\alpha}]}{\text{Re}[\mathcal{E}_{\alpha}]}}\right)$
as a function of $\gamma$. In absence of system-lead coupling
$\theta_{\mathcal{E}_{\alpha}}(\gamma=0)=\frac{2\pi}{3}$ for the specific case
of DTC. Fig. 3 shows that $\theta_{\mathcal{E}_{\alpha}}(\gamma)$ remains
locked at $\frac{2\pi}{3}$, and deviates only about $0.5\%$ even at the
largest value of $\gamma=\omega$ we have considered. We have checked that, the
deviation is further related to the numerical accuracy corresponding to time
of $dt$ of the evolution. In an otherwise perfect numerical calculation
($dt\rightarrow 0$) it would be negligibly small. This frequency locking will
further be seen in the oscillation of the spin-current.

Figure 4: discrete time crystal (DTC) and discrete time quasi-crystal (DTQC)
oscillations. Time evolution of $\langle S^{y}_{j}(t)\rangle$ [$j=1$ (blue
solid lines),$j=2$ (orange dot-dashed lines),$j=3$ (green dashed lines)] as a
function of the dimensionless time $t/T$ showing the appearance of DTC in the
long-time limit for three values of system-lead couplings, (a)
$\gamma=10^{-5}\omega$, (b) $\gamma=10^{-3}\omega$, and (c)
$\gamma=\omega/50$. Black dotted line represents $\cos(\frac{2\pi t}{T})$, the
driving period. (d), (e), (f): Discrete Fourier transform (DTF)
$\mathcal{F}(\langle S^{y}_{3}\rangle)$ of $\langle S^{y}_{3}(t)\rangle$ as a
function of the frequency domain variable $f/\omega$ corresponding to (a), (b)
and (c), respectively, showing the $\lambda=2\omega/3$. Time evolution of
$\langle S^{y}_{j}(t)\rangle$ ($j=1,2,3$) as a function of the dimensionless
time $t/T$ showing the appearance of DTQC in the long-time limit for three
values of system-lead couplings, (g) $\gamma=10^{-5}\omega$, (h)
$\gamma=10^{-3}\omega$, and (i) $\gamma=\omega/50$. (j), (k), (l): DFT of
$\langle S^{y}_{3}(t)\rangle$ as a function of the frequency domain variable
$f/\omega$ corresponding to (g), (h) and (i), respectively, showing the
$\lambda=(\sqrt{2}-1)\omega$. There exists another frequency at an integer
multiple of the DTC (and DTQC) frequency in the form of a tiny peak in DTF
above $\omega$.

### Synchronized long-lived DTC and DTQC

With the notion of the density matrix corresponding to the DTC (and DTQC) in
terms of the decaying coherent states, we numerically demonstrate the role of
system-lead coupling in bringing a synchronized stable and meta-stable DTC and
DTQC fingerprinted in the oscillation of $\langle S_{j}^{y}(t)\rangle$
($j=1,2,3$ is the site index) and the spin current. Following the standard
notions Mari  _et al._ (2013); Tindall  _et al._ (2020), by synchronization we
mean that the oscillation of $\langle S_{j}^{y}(t)\rangle$ corresponding to
each site is locked to the same frequency and phase, and exhibit the same
magnitude independent of the specific value of any microscopic parameters.

We start from an initial density matrix corresponding to a half filled thermal
state at 77K, where the system is placed under a (suitably strong) constant
static magnetic field $B_{z}$ in z-direction. Such a configuration is purely
chosen as a convenience because it is easy to create in a quantum-dot array.
Subsequently, the system is quenched to a state of $B_{z}=0$, and at the same
time applied with a circularly polarized magnetic field $V(t)$. However, a
random initial density matrix also shows the same results provided it has
substantial overlap with the decaying coherent states $\rho_{mn}$.

Figs. 4 (a), (b) and (c) show the time evolutions of $\langle
S_{j}^{y}\rangle$ in the DTC behavior for $B=\frac{4}{3}\omega$ such that
$|\mathbf{h}|=\frac{2}{3}\omega$ (note the modulo operation). Three
representative values of $\gamma$ are considered here, corresponding to
$\gamma=10^{-5}\omega$ (very weak system-lead coupling) in Fig. 4 (a),
$\gamma=10^{-3}\omega$ (moderate system-lead coupling) in Fig. 4 (b), and
$\gamma=\omega/50$ (strong system-lead coupling) in Fig. 4 (c), respectively.
A system-lead coupling, $10^{-5}\omega\leq\gamma\leq 10^{-3}\omega$ correspond
to values in the range of $62~{}KHz\leq\gamma\leq 6.2~{}KHz$, which are a
standard in transport experiments with quantum-dot-array set-up Gustavsson
_et al._ (2007).

After an initial relaxation dynamics characterized by the value of the
dephasing strength $\Gamma$ ($0.1t_{hop}$ in our case), $\langle
S_{j}^{y}\rangle$ exhibits a stable oscillation with a period $3T$ for weak
system-lead coupling, see Fig. 4 (a). Moreover, the dynamics of $\langle
S_{j}^{y}\rangle$ corresponding to all the three sites are synchronized due to
the spatial translational symmetry in the density matrix corresponding to the
FSS and the other coherent states Buča  _et al._ (2019); Tindall  _et al._
(2020); Chinzei and Ikeda (2020). Figs. 4 (d) shows the corresponding
normalized discrete Fourier transform (DFT) exhibiting a peak at
$\frac{2}{3}\omega$ commensurately with the driving frequency $\omega$. The
secondary peak at $\frac{8}{3}\omega$ is an integer multiple of the DTC peak.

In Fig. 4 (b), we find that after the initial relaxation $\langle
S_{j}^{y}\rangle$ oscillates with decaying magnitude leading to a meta-stable
oscillation over a considerable time duration (at-least a few tens of driving
periods) before eventually decay to a stationary state with no oscillation.
The corresponding DFT is shown in Fig. 4 (e). This decay occurs in a time
scale of $\gamma^{-1}$. This is a manifestation of the fact that for
$\gamma=10^{-3}\omega=0.063~{}\text{MHz}$ the DTC eigenvalue of
$\mathcal{U}_{F}$ starts to deviate from the peripheral spectrum, see Fig. 2
(c).

Fig. 4 (c) further indicates that a stronger damping due to larger values of
$\gamma$ leads to a broadened peak in DFT indicating superposition of many
frequencies around $\frac{2}{3}\omega$ and $\omega$ signaling a noisy
oscillation. This results in broadened DFT peaks seen in Fig. 4 (f). This is a
manifestation of the fact that $\gamma$ acts as a detuning parameter that
force $\text{Im}[\mathcal{E}_{DTC}]$ to deviate from its $\gamma=0$ value, see
Fig. 2 (c).

A larger system corresponding to N=5 also exhibits the exact same oscillation
in $\langle S^{y}_{j}(t)\rangle$ with $3T$ time-period of the DTC, see
Supplementary Note 4.

Although dephasing has no effect on the DTC time period, it determines the
time scale over which spin-oscillations corresponding to each site get
completely synchronized with the other sites. For a given system-lead coupling
the first and the last sites are always synchronized irrespective of the value
of the dephasing rate. However, all the sites that are not connected to the
external leads synchronize on a time scale $\sim\Gamma^{-1}$ with the ones
connected to the leads (see Supplementary Note 5). This is true irrespective
of the system size, as shown in Supplementary Figure 2 for both four and five
sites systems.

The above conclusions hold true also for the DTQC shown for
$\gamma=10^{-5}\omega$ in Fig. 4 (g), $\gamma=10^{-3}\omega$ in Fig. 4 (h),
and $\gamma=\omega/50$ in Fig. 4 (i), respectively. Fig. 4 (j)-(l) show the
corresponding normalized DFTs exhibiting peaks at $f=(\sqrt{2}-1)\omega$
incommensurate with the driving frequency $\omega$. It is worthwhile to point
out that if $B$ and $\omega$ are of the same orders of magnitude or at-least
$B>>\omega$, then the DTC/DQTC can be observed. In the case of $B<<\omega$,
the coherence frequency scale $\lambda\approx\omega$ and the system follows
the conventional Floquet response oscillating with the period of the drive,
and no time crystallinity can be observed.

Figure 5: Limit cycles. Blue lines are trajectories of the time evolution in
$\langle S_{3}^{x}\rangle-\langle S_{3}^{y}\rangle-\langle S_{3}^{z}\rangle$
space, (a) and (b) for stable discrete time crystal (DTC) and discrete time
quasi-crystal (DTQC), (c) and (d) meta-stable DTC and DTQC, respectively.
Orange, green and red lines are projections on $\langle
S_{3}^{x}\rangle-\langle S_{3}^{y}\rangle$, $\langle S_{3}^{x}\rangle-\langle
S_{3}^{z}\rangle$ and $\langle S_{3}^{y}\rangle-\langle S_{3}^{z}\rangle$
planes, respectively. Figure 6: Spin-currents from discrete time crystal (DTC)
and time quasi-crystal (DTQC): Time evolution of the spin currents $J_{s}(t)$,
in pico-ampere (pA), as a function of the dimensionless time $t/T$ showing the
signature of DTC in the long-time limit for three values of system-lead
couplings (a) $\gamma=10^{-5}\omega$ (very weak system-lead coupling), (b)
$\gamma=10^{-3}\omega$ (moderate system-lead coupling), and (c)
$\gamma=\omega/50$ (strong system-lead coupling). (d), (e), (f) correspond to
discrete Fourier transform (DFT) $\mathcal{F}[J_{s}(t)]$ of the spin current
$J_{s}(t)$ as a function of the frequency domain variable $f/\omega$
corresponding to (a), (b), and (c) respectively, showing the DTC peak at
$f=\frac{5\omega}{3}$. Time evolution of the spin currents $J_{s}(t)$, in pA,
as a function of the dimensionless time $t/T$ showing the signature of DTQC in
the long-time limit for three values of system-lead couplings (g)
$\gamma=10^{-5}\omega$ (very weak system-lead coupling), (h)
$\gamma=10^{-3}\omega$ (moderate system-lead coupling), and (i)
$\gamma=\omega/50$ (strong system-lead coupling). (j), (k), (l) correspond to
DFT $\mathcal{F}[J_{s}(t)]$ of the spin current as a function of the frequency
domain variable $f/\omega$ corresponding to (g), (h), and (i), respectively,
showing the DTQC peak at $f=\sqrt{2}\omega$.

### Limit-cycles

Although the previous section showed a DTC behavior in $\langle
S^{y}(t)\rangle$, we point out that $\langle S^{y}(t)\rangle$ is not easily
observed. A connection of the DTC behavior in $\langle S^{y}(t)\rangle$ to a
more accessible quantity such as $\langle S^{z}(t)\rangle$, is provided
through the limit cycle analysis. A limit cycle is a closed trajectory in the
phase-space Pikovsky  _et al._ (2001), in our case the phase-space of the
dynamical variables, viz., $\langle S_{j}^{x}\rangle$, $\langle
S_{j}^{y}\rangle$ and $\langle S_{j}^{z}\rangle$. Fig. 5 shows the limit cycle
oscillations for both stable and meta-stable DTC and DTQC, respectively. To
obtain the time evolution we solve (6) [see Methods section] and calculate
$\langle S_{j}^{\alpha}(t)\rangle=Tr[S_{j}^{\alpha}\rho(t)]$ for $j=x,y,z$.
The limit cycles are plotted for a time duration of fifty driving periods, for
$\gamma=(10^{-5}\omega,10^{-3}\omega)$ [Fig.5(a,c)] and
$B=(\frac{4}{3}\omega,\omega)$ [Fig.5(b,d)]. While Figs. 5 (a) and (b) show a
stable limit cycle characterized by a well defined closed path, 5 (c) and (d)
show that the limit cycle shrinks ( but very slowly) over fifty driving
periods. The limit cycle oscillations indicate that although $S_{j}^{x,(y)}$
respect weak local FDS, $S_{j}^{z}$ also oscillates, providing the means of
experimentally extracting a measurable transport signature.

### Transport signature– oscillating spin-current

Spin current (operator) is defined as,

$\hat{J}_{s}=e\gamma(n_{N,\uparrow}-n_{N,\downarrow})=2e\gamma S_{N}^{z},$ (3)

where $e$ is electronic charge, $n_{N,\sigma}$ is the density operator
corresponding to the spin $\sigma$ at the extraction site connected to the
right lead, see Supplementary Note 6. Fig. 6 plots the time evolution of the
expectation value of the spin-current, defined above, viz.,
$J_{s}(t)=\text{Tr}[\hat{J}_{s}\rho(t)]$ and its DFT. In the case of DTC
corresponding to Figs. 6 (a) and (b), a stable and a meta-stable DTC are seen
for weak and moderate system-lead couplings, respectively. Importantly enough,
the DFT of the corresponding oscillations show only one sharp peak at
$f=|\mathbf{h}|$ for both stable and meta-stable DTC, see Figs. 6 (d) and (e),
respectively. This is a direct measurable signature of the DTC in the sense
that if the DFT from the spin-current shows a peak at a rational fraction of
the driving frequency, i.e., $f/\omega=q/p$ one concludes there exists a DTC
with a period $pT$. As usual, a strong system-lead coupling destroys the DTC
as seen in Fig. 6 (c) and indicated by the broadened peak in the DFT of the
spin current in Fig. 6 (f). Figs. 6 (g), (h) and (i) show the oscillation of
the spin current corresponding to DTQC for weak, moderate, and strong system-
lead couplings, respectively. Likewise, the sharp peak in DFT of the spin
current in Figs. 6 (j), (k) and (l) directly provides the exact value of the
time period of the DTQC at $f=\sqrt{2}\omega$. The oscillation of spin-current
remains exactly the same for the 5-site system, which is apparent from the
fact that oscillations in $x-$ and $y-$ components of the spins are exactly
the same as that of the 3-site system.

## Conclusions

We discover, in an experimentally relevant paradigmatic model, viz., Floquet
Fermi-Hubbard chain connected to electrodes, DTC and DTQC are manifested
through the oscillations of the spin-current. We show that a unique form of
“weak local Floquet dynamical” symmetry emerges in the long time limit which
preserves the DTC and the DTQC. The amplitude of the DTC oscillation remain
appreciable if the time scale corresponding to the system-lead coupling is
much larger than the time period of the drive, viz., $\gamma^{-1}\gg T$. Both
DTC and DTQC survive in the long time limit (over 100 periods of external
drive) in line with recent experiments performing optical detection of DTCs in
closed systems Kyprianidis  _et al._ (2021); Mi  _et al._ (2021). Due to the
“weak local Floquet dynamical symmetry,” a transient/meta-stable manifold
emerges which is distinct from the FSS because, with increasing system-
electrode coupling, it never coalesces to FSS. DTC and DTQC decay linearly
with system-lead coupling strength with slope of the decay being twice the
driving period. Our findings thus highlight that by fine-tuning the system-
electrode coupling (a form of dissipation engineering) one can obtain a
sustained DTC behavior and also allow charge flow through the system allowing
measurement of DTC behavior through electrical transport. Put simply, the
dynamical symmetry provides the mechanism of DTC/DTQC and meta-stability due
to charge transport provides the needed decay channel that aid to measurement
Volovik (2013).

Our results show that although the system-lead coupling induces a decay in the
oscillation amplitude of DTC, the corresponding sub-harmonic frequency remains
locked during the decay, which means that the signature of time-crystallinity
can be measured even as it decays. This unfolds an underlying mathematical
structure of weak-local Floquet dynamical symmetry. In the long time limit,
the system relaxes to a sub-space of of the full super-Hilbert space which is
rather non-trivial and can not be a-priori anticipated. Indeed, there can be
systems where various effects of environments, such as system-lead coupling,
may also lead to the destruction of time-crystal in terms of its oscillation
frequency, in which case no signature of the time-crystal can be observed.

Given the pure time-dependent spin current is detectable in the inverse spin-
Hall effect Wei  _et al._ (2014), our predictions are experimentally testable
in the presently available quantum-dot array set-up where a Fermi-Hubbard
chain has recently been realized Barthelemy and Vandersypen (2013);
Kouwenhoven  _et al._ (2001); Hensgens  _et al._ (2017); Mills  _et al._
(2019); Zajac  _et al._ (2018); Han  _et al._ (2020). Ultra-cold quantum gases
also provide another promising route for experimental study the transport
properties of one dimensional many-body systems Gross and Bloch (2017);
Brantut  _et al._ (2012); Chien  _et al._ (2015). In optical lattices,
periodically driven Fermi-Hubbard model is rather routinely analyzed Esslinger
(2010), and with the existence of a cold-atom analogue of mesoscopic conductor
Brantut  _et al._ (2012); Chien  _et al._ (2015) our findings can be verified
in optical lattices too.

## Methods

### Hamiltonian

The Hamiltonian corresponding to periodically driven Fermi-Hubbard chain, see
Fig. 1, comprising of $N$ sites is given by
$\mathcal{H}(t)=\mathcal{H}_{0}+H_{ext}(t)$ where Chinzei and Ikeda (2020);
Hensgens  _et al._ (2017)

$\displaystyle\mathcal{H}_{0}$ $\displaystyle=$ $\displaystyle-
t_{hop}\sum_{\begin{subarray}{c}j=1,\\\
\sigma=\uparrow,\downarrow\end{subarray}}^{N-1}(c_{j,\sigma}^{\dagger}c_{j+1,\sigma}+c_{j+1,\sigma}^{\dagger}c_{j,\sigma})+\sum_{j=1}^{N}\frac{U}{2}n_{j}n_{j}$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{N-1}\frac{K}{2}n_{j}n_{j+1};~{}\text{with}~{}n_{j}=\sum_{\sigma=\uparrow,\downarrow}n_{j,\sigma},$
$\displaystyle H_{ext}(t)$ $\displaystyle=$ $\displaystyle
B\sum_{j=1}^{N}(S^{x}_{j}\cos\omega t+S^{y}_{j}\sin\omega t).$ (4)

In eq. (Hamiltonian), $t_{hop},~{}U,~{}\text{and}~{}K$ are the nearest-
neighbor hopping, onsite electron-electron, and the nearest-neighbor electron-
electron interaction strengths, respectively; $B$ is the magnitude of the
external magnetic field induced by a circularly polarized laser of frequency
$\omega=\frac{2\pi}{T}$ leading to $\mathcal{H}(t)=\mathcal{H}(t+T)$.

### Lindblad equation

The system is then subjected to onsite dephasing, represented by the Lindblad
operators, $V_{D,j}=\sum_{\sigma}\sqrt{\Gamma}n_{j,\sigma}$ and connected to
leads,
$V_{L}=\sqrt{\gamma_{L}}\sum_{\sigma=\uparrow,\downarrow}c_{1,\sigma}^{\dagger}$
and $V_{R}=\sqrt{\gamma_{R}}\sum_{\sigma=\uparrow,\downarrow}c_{N,\sigma}$,
where $\gamma_{L(R)}$ is the system- left(right) lead coupling which we take
to be equal. The choice of the lead operators mimics the infinite bias
condition. The non-unitary dynamics of the system (and the corresponding
density matrix $\rho$) is governed by the Floquet-Lindblad equation (within
Born-Markov approximation) Lindblad (1976); Gorini  _et al._ (1976); Breuer
and Petruccione (2004); Purkayastha  _et al._ (2016),

$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{t}[\rho]=-i[\mathcal{H}(t),\rho]$ (5)
$\displaystyle+$ $\displaystyle\sum_{\begin{subarray}{c}\mu=D,\\\
L,R\end{subarray}}\left(V_{\mu}\rho
V_{\mu}^{\dagger}-\frac{1}{2}\\{V_{\mu}^{\dagger}V_{\mu},\rho\\}\right),$

(we set $\hbar=1$). The periodicity of the Hamiltonian guarantees the
periodicity of the Liouvillian $\mathcal{L}_{t}=\mathcal{L}_{t+T}$.

The density matrix, obeying the Lindblad equation therefore, is an
$2^{2N}\times 2^{2N}$ positive definite matrix with trace one. The standard
practice of solving the (5) is to vectorize the $\rho$ into a $2^{4N}$
dimensional vector $|\rho\rangle\rangle$. This results in a reformulation of
the Lindblad equation into a $2^{4N}$ dimensional Linear ODE with time-
dependent coefficients ,

$\dfrac{|\rho(t)\rangle\rangle}{dt}=\mathcal{L}_{t}|\rho(t)\rangle\rangle,$
(6)

$\mathcal{L}_{t}$ being the Lindblad super-operator of dimension $2^{4N}\times
2^{4N}$. The infinitesimal time evolution is governed by $|\rho(t+\Delta
t)\rangle\rangle=e^{\mathcal{L}_{t}\Delta t}|\rho(t)\rangle\rangle$.

Alternatively, the Floquet-Lindblad form for any operator $A(t)$ is given by,

$\dfrac{dA(t)}{dt}=i[\mathcal{H}(t),A]+\frac{1}{2}\sum_{\mu}\left(V_{\mu}^{\dagger}\left[A,V_{\mu}\right]+\left[V_{\mu}^{\dagger},A\right]V_{\mu}\right),$
(7)

(notice $\hbar=1$) where the periodic Hamiltonian $\mathcal{H}(t)$ is given by
(Hamiltonian).

### Observables

The oscillations of the spin-components and the spin-current are obtained from
the formula, $\langle A(t)\rangle=\text{Tr}[A\rho(t)]$, where $A$ represents
the operators whose expectation values are plotted in Figs. 4 and 6. The
$2^{2N}\times 2^{2N}$ time -dependent density matrix $\rho(t)$ is obtained by
reshaping $2^{4N}$ dimensional vector $|\rho(t)\rangle\rangle$. Alternatively,
we also use Runge-Kutta method for larger system sizes, N=4 and 5. This way of
solving the Lindblad equation can be performed in $2^{2N}\times 2^{2N}$
dimensional space.

## Data availability

All relevant data are available from the corresponding author upon reasonable
request.

## Author contribution

SS performed all the analytical and numerical calculations. SS and YD
discussed the results and wrote the manuscript.

## Competing interests

The authors declare no competing interests.

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# Diophantine Equation with Balancing-like Sequences Associated to the Pillai-
Tijdeman-type Problem

Bijan Kumar Patel and Prashant Tiwari

###### Abstract

Let $\\{x_{n}\\}_{n\geq 0}$ be the balancing-like sequence defined by
$x_{n+1}=Ax_{n}-x_{n-1}$, for $A>2$, where $x_{0}=0$ and $x_{1}=1$. In this
paper, we demonstrate how to find all the solutions of the Diophantine
equation,
$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{4}x_{n_{4}}+C_{5}x_{n_{5}}+C_{6}x_{n_{6}}$,
in fixed integer $A\geq 3$, $n_{1}>n_{2}>n_{3}\geq 0,n_{4}>n_{5}>n_{6}\geq 0,$
and $C_{1}x_{n_{1}}\neq C_{4}x_{n_{4}}$, where
$C_{1},C_{2},C_{3},C_{4},C_{5},C_{6}$ are given integers such that
$C_{1}C_{2}C_{3}\neq 0$.

Keywords: Balancing sequence; Balancing-like sequence; Diophantine equations.
2010 Mathematics Subject Classification: 11B39; 11B83; 11D61.

##  1 Introduction

A balancing number $B$ is a natural number satisfying the Diophantine equation

$\sum_{j=1}^{B-1}j=\sum_{k=B+1}^{R}k,$

for some natural number $R$ [1]. If $B$ is a balancing number, then $8B^{2}+1$
is a perfect square. The $n$-th balancing number is denoted by $B_{n}$ and
satisfy the binary recurrence $B_{n+1}=6B_{n}-B_{n-1}$ with initial values
$(B_{0},B_{1})=(0,1).$ The balancing sequence has been studied extensively and
generalized in many ways [4, 5]. As a generalization of the balancing
sequence, Panda and Rout [4] introduced a family of binary recurrences defined
by

$x_{n+1}=Ax_{n}-x_{n-1},~{}~{}x_{0}=0,~{}x_{1}=1,$

for any fixed integer $A>2$. Afterwards, these sequences were called as
balancing-like sequences $\\{x_{n}\\}_{n\geq 0}$ since the particular case
corresponding to A = 6 coincides with the balancing sequence. When $A=2$, the
binary sequence $x_{n+1}=2x_{n}-x_{n-1}$ with $x_{0}=0,x_{1}=1$ generates the
natural numbers. The sequence $\\{x_{n}\\}_{n\geq 0}$ behaves like the
sequence of natural numbers, and for this reason, Khan and Kwong [3] called
these sequences as generalized natural number sequences. For each $n$,
$Dx_{n}^{2}+1$ is a perfect rational square, where $D=\frac{A^{2}-4}{4}$, is a
perfect square and its positive square root is called a Lucas-balancing-like
number [4]. They have also noted that all the balancing-like sequences are
strong divisibility sequences. Further, different associate sequences of a
balancing-like sequence has been studied by Panda and Pradhan [5]. The Binet
form of the balancing-like sequence $\\{x_{n}\\}_{n\geq 0}$ is given by

$x_{n}=\frac{\gamma^{n}-\delta^{n}}{\sqrt{A^{2}-4}},~{}~{}n=1,2,\dots,$

where $\gamma:=\frac{A+\sqrt{A^{2}-4}}{2}$ and
$\delta:=\frac{A-\sqrt{A^{2}-4}}{2}$ are the roots of the characteristic
equation $X^{2}-AX+1=0$. It is easy to see that $\gamma=\delta^{-1}.$
Balancing-like sequence satisfies the following divisibility property.

######  Lemma 1.1 (Theorem 2.8, [4]).

For all $m,n\in\mathbb{N}$, $\gcd(x_{m},x_{n})=x_{\gcd(m,n)}$.

Furthermore, one can prove by induction that the inequality

$\gamma^{n-2}\leq x_{n}\leq\gamma^{n-1}$ (1.1)

holds for all positive integers $n$.

Sahukar and Panda [6] dealt with the Brocard-Ramanujan-type equations
$x_{n_{1}}x_{n_{2}}\dots x_{n_{k}}\pm 1=x_{m}$ or $y_{m}$ or $y_{m}^{2}$ where
$\\{x_{n}\\}_{n\geq 0}$ and $\\{y_{m}\\}_{m\geq 0}$ are either balancing-like
sequences or associated balancing-like sequences. Subsequently they [7] showed
that each balancing-like sequence has at most three terms which are one away
from perfect squares.

Shorey and Tijdeman [8] proved under suitable conditions, that the equation
$Ax^{m}+By^{m}=Cx^{n}+Dy^{n}$ implies that max$\\{n,m\\}$ is bounded by a
computable constant depending only on $A,B,C,D$, where $A,B,C,D,m,n>0$.
Recently, Ddamulira et al. [2] studied a variation of the equation
$Ax^{m}+By^{m}=Cx^{n}+Dy^{n}$ with the terms of the Lucas sequence
$\\{U_{n}\\}_{n\geq 0}$. They showed that there are two types of solutions of
the Diophantine equation $AU_{n}+BU_{m}=CU_{n_{1}}+DU_{m_{1}}$, sporadic
solutions and parametric solutions.

Motivated by the above works, we study the Diophantine equation

$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{4}x_{n_{4}}+C_{5}x_{n_{5}}+C_{6}x_{n_{6}}$
(1.2)

with $n_{1}>n_{2}>n_{3}\geq 0,n_{4}>n_{5}>n_{6}\geq 0,$ and
$C_{1}x_{n_{1}}\neq C_{4}x_{n_{4}}$, where $\\{x_{n}\\}_{n\geq 0}$ is a
balancing-like sequence and $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6}$ are given
integers such that $C_{1}C_{2}C_{3}\neq 0$. More precisely, our main result is
the following.

######  Theorem 1.2.

Let $X:=\max\\{|C_{1}|,|C_{2}|,|C_{3}|,|C_{4}|,|C_{5}|,|C_{6}|\\}$ and
$\psi:=(3+\sqrt{5})/2$ be the smallest possible $\gamma$. Relabeling the
variables $(n_{1},n_{2},n_{3},n_{4},n_{5},n_{6})$ as
$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6}),$ where $m_{1}\geq m_{2}\geq m_{3}\geq
m_{4}\geq m_{5}\geq m_{6}$. If $n_{1}=n_{4}$, we rewrite the Diophantine
equation (1.2) as

$(C_{1}-C_{4})x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{5}x_{n_{5}}+C_{6}x_{n_{6}},$

and change $(C_{1},C_{2},C_{3},C_{4},C_{5},C_{6})$ to
$(C_{1}-C_{4},C_{2},C_{3},C_{5},C_{6},0)$. Thus, $n_{1}>n_{4}$. Furthermore,
we change the sign of some of the coefficients
$(C_{1},C_{2},C_{3},C_{4},C_{5},C_{6})$ so that the Diophantine equation (1.2)
becomes

$A_{1}x_{m_{1}}+A_{2}x_{m_{2}}+A_{3}x_{m_{3}}+A_{4}x_{m_{4}}+A_{5}x_{m_{5}}+A_{6}x_{m_{6}}=0.$
(1.3)

Assume $A\leq 308X$ for any fixed integer $A>2$ of the characteristic equation
$X^{2}-AX+1=0$. Then, the solutions of the Diophantine equation (1.3) are of
two types:

  1. (i)

Sporadic ones. These are finitely many and they satisfy:

$\displaystyle
m_{6}\leq\frac{\log(12X)}{\log\psi},m_{5}\leq\frac{\log(1000X^{3})}{\log\psi},m_{4}\leq\frac{\log(123000X^{5})}{\log\psi},$
$\displaystyle
m_{3}\leq\frac{\log(17700000X^{7})}{\log\psi},m_{2}\leq\frac{\log(4530000000X^{9})}{\log\psi},m_{1}\leq\frac{\log(99660000000X^{10})}{\log\psi}.$

  2. (ii)

Parametric ones. These are of one of the two forms:

$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})=(m_{5}+l,m_{5}+k,m_{5}+j,m_{5}+i,m_{5},0),$

where

$\displaystyle
l\leq\frac{\log(104000000X^{7})}{\log\psi},~{}~{}~{}k\leq\frac{\log(4720000X^{6})}{\log\psi},$
$\displaystyle
j\leq\frac{\log(18500X^{4})}{\log\psi}~{}~{}~{}\text{and}~{}~{}~{}i\leq\frac{\log(130X^{2})}{\log\psi},$

and $\gamma$ is a root of
$A_{1}X^{l}+A_{2}X^{k}+A_{3}X^{j}+A_{4}X^{i}+A_{5}=0$, or of the form

$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})=(m_{6}+m,m_{6}+l,m_{6}+k,m_{6}+j,m_{6}+i,m_{6}),$

where

$\displaystyle m$
$\displaystyle\leq\frac{\log(8305000000X^{9})}{\log\psi},~{}~{}~{}l\leq\frac{\log(377500000X^{8})}{\log\psi},~{}~{}~{}k\leq\frac{\log(1485000X^{6})}{\log\psi},$
$\displaystyle
j\leq\frac{\log(10300X^{4})}{\log\psi}~{}~{}~{}\text{and}~{}~{}~{}i\leq\frac{\log(80X^{2})}{\log\psi},$

and $\gamma$ is a root of

$A_{1}X^{m}+A_{2}X^{l}+A_{3}X^{k}+A_{4}X^{j}+A_{5}X^{i}+A_{6}=0.$

##  2 Main Result

To achieve the objective of this paper, we need the following result.

######  Lemma 2.1.

Let $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6}$ are given integers such that
$C_{1}C_{2}C_{3}\neq 0$. If the Diophantine equation
$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{4}x_{n_{4}}+C_{5}x_{n_{5}}+C_{6}x_{n_{6}}$
with $n_{1}>n_{2}>n_{3}\geq 0,n_{4}>n_{5}>n_{6}\geq 0,C_{1}x_{n_{1}}\neq
C_{4}x_{n_{4}}$ holds, then $A<308X$.

###### Proof.

Case I If $C_{3}=C_{6}=0$, then
$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}=C_{4}x_{n_{4}}+C_{5}x_{n_{5}}$ for any
$n_{3},n_{6}\geq 0$. We have the following subcases for it.

Subcase 1 If $C_{4}=C_{5}=0$, then $C_{1}x_{n_{1}}=-C_{2}x_{n_{2}}$ for any
$n_{4}>n_{5}\geq 0$. As we have given $C_{1}x_{n_{1}}\neq C_{4}x_{n_{4}}$ and
$C_{1}C_{2}\neq 0$, using these one can get $n_{1}$ and $n_{2}$ both are non-
zero. Thus from Lemma 1.1 and $n_{2}\neq 0$, we have $\frac{x_{n_{1}}}{x_{d}}$
divides $C_{2}$, where $d:=\gcd(n_{1},n_{2}).$ Write $n_{1}=:ld$, where $l\geq
2$. If $d=1$, then $\frac{x_{n_{1}}}{x_{d}}=\frac{x_{l}}{x_{1}}=x_{l}\geq
x_{2}=A$, so $A\leq X$. For $d\geq 2$, we have
$\frac{x_{n_{1}}}{x_{d}}=\frac{\gamma^{ld}-\delta^{ld}}{\gamma^{d}-\delta^{d}}$.
Now, we will show that

$\frac{\gamma^{ld}-\delta^{ld}}{\gamma^{d}-\delta^{d}}>\gamma,$

which is equivalent to $\gamma^{ld}-\gamma^{d+1}>\delta^{ld}-\delta^{d-1}$.
Since $d\geq 2$ and $|\delta^{d-1}|<1$, therefore it is sufficient to show
that $\gamma^{2d}-\gamma^{d+1}>29$. The left–hand side is

$\displaystyle\gamma^{2d}-\gamma^{d+1}$
$\displaystyle=\gamma^{d+1}(\gamma^{d-1}-1)$
$\displaystyle\geq\gamma^{d+1}(\gamma-1).$

The smallest possible $\gamma$ is $\psi=\frac{3+\sqrt{5}}{2}$ (for $A=3$) and
$\psi^{d+1}(\psi-1)\geq\psi^{3}(\psi-1)>29$. Hence,
$\gamma<\frac{x_{ld}}{x_{d}}\leq X$, which gives $A=\gamma+\delta<\gamma<X$.
Using the inequality (1.1) we have $x_{n_{1}}\geq\gamma^{n_{1}-2}$ and
$x_{d}\leq\gamma^{d-1}$, so
$\frac{x_{n_{1}}}{x_{d}}\geq\frac{\gamma^{n_{1}-2}}{\gamma^{d-1}}=\gamma^{n_{1}-d-1}$.
Since $d$ is a proper divisor of $n_{1}$ and $d\geq 2$, so
$d\leq\frac{n_{1}}{2}$, therefore we have
$\frac{x_{n_{1}}}{x_{d}}\geq\gamma^{\frac{n_{1}}{2}-1}$. Since
$\frac{x_{n_{1}}}{x_{d}}$ divides $C_{2}$, we get

$\gamma^{\frac{n_{1}}{2}-1}\leq|C_{2}|\leq X.$

Taking the logarithm of both sides of the above equation leads to

$n_{1}\leq 2+2\frac{\log X}{\log\gamma}.$

Since $\gamma\geq\psi$, we have

$0<n_{2}<n_{1}\leq 2+2\frac{\log X}{\log\psi}.$ (2.4)

Subcase 2 If one of $C_{4},C_{5}$ is non-zero and the other is zero.
Therefore, we can assume that $C_{4}\neq 0$ and $n_{4}\neq 0$. Thus, if
$C_{5}=0$, then

$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}=C_{4}x_{n_{4}}.$

If $n_{4}=n_{1}$, then $(C_{1}-C_{4})x_{n_{1}}=-C_{2}x_{n_{2}}$, so from
subcase 1, we get the bound $A\leq X$. If $n_{4}\neq n_{1}$ and $n_{2}=0$,
then again from subcase 1, we get the bound $A\leq X$. If $n_{4}\neq n_{1}$
and $n_{2}\neq 0$, then $C_{1}x_{n_{1}}+C_{2}x_{n_{2}}=C_{4}x_{n_{4}}$. In
this situation we replace
$(C_{1},C_{2},C_{4},C_{5})\longrightarrow(C_{1},C_{2},C_{4},0)$ and in this
case also we get the same bound which is $A\leq X$.

Subcase 3 If both $C_{4}$ and $C_{5}$ are non-zero, then

$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}=C_{4}x_{n_{4}}+C_{5}x_{n_{5}}.$

If $n_{1}=n_{4}$, then $(C_{1}-C_{4})x_{n_{1}}+C_{2}x_{n_{2}}=C_{5}x_{n_{5}}$.
If $n_{2}=0$ or $n_{5}=0$, then again from subcase 1, we get the same bound
$A\leq X$, otherwise we replace
$(C_{1},C_{2},C_{4},C_{5})\longrightarrow(C_{1}-C_{4},C_{2},C_{5},0)$. The
only effect is that $X$ is replaced with $2X$. If $n_{1}\neq n_{4}$, then
switch $C_{1}$ with $C_{4}$. If needed, assume $n_{1}=\max\\{n_{4},n_{1}\\}$,
then $n_{1}>n_{4}$. We relabel our indices $(n_{1},n_{2},n_{4},n_{5})$ as
$(m_{1},m_{2},m_{3},m_{4})$ where $m_{1}>m_{2}\geq m_{3}\geq m_{4}$, and the
coefficients $C_{1},C_{2},C_{4},C_{5}$ as $A_{1},A_{2},A_{3},A_{4}$ and change
signs to at most a couple of them so that our equation is now

$A_{1}x_{m_{1}}+A_{2}x_{m_{2}}+A_{3}x_{m_{3}}+A_{4}x_{m_{4}}=0.$ (2.5)

Using Binet formula of $\\{x_{n}\\}_{n\geq 0}$, we have

$|A_{1}|\gamma^{m_{1}}=|-A_{2}(\gamma^{m_{2}}-\delta^{m_{2}})-A_{3}(\gamma^{m_{3}}-\delta^{m_{3}})-A_{4}(\gamma^{m_{4}}-\delta^{m_{4}})-A_{1}\delta^{m_{1}}|<7X\gamma^{m_{2}},$

therefore

$\gamma^{m_{1}-m_{2}}<7X.$ (2.6)

Thus, since $m_{1}>m_{2}$, we get that $A<\gamma\leq\gamma^{m_{1}-m_{2}}<7X$.
Recalling that we might have to replace $X$ with $2X$, we get the final bound
in this case $A<14X$.

Case II If one of $C_{5},C_{6}$ is non-zero and the other is zero. We assume
that $C_{5}\neq 0$ and $n_{5}\neq 0$. Thus, if $C_{6}=0$, then

$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{4}x_{n_{4}}+C_{5}x_{n_{5}}.$

If $n_{1}=n_{4}$, then
$(C_{1}-C_{4})x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{5}x_{n_{5}}$. So
using the Case I, we get the bound $A<14X$. Recalling that we might have to
replace $X$ with $2X$, we get the final bound in this case $A<28X$. If
$n_{1}\neq n_{4}$, then switch $C_{1}$ with $C_{4}$. If needed, we may assume
that $n_{1}=\max\\{n_{4},n_{1}\\}$, therefore $n_{1}>n_{4}$. We relabel our
indices $(n_{1},n_{2},n_{3},n_{4},n_{5})$ as $(m_{1},m_{2},m_{3},m_{4},m_{5})$
where $m_{1}>m_{2}>m_{3}\geq m_{4}\geq m_{5}$, and the coefficients
$C_{1},C_{2},C_{3},C_{4},C_{5}$ as $A_{1},A_{2},A_{3},A_{4},A_{5}$ and change
signs to at most a couple of them so that our equation is now

$A_{1}x_{m_{1}}+A_{2}x_{m_{2}}+A_{3}x_{m_{3}}+A_{4}x_{m_{4}}+A_{5}x_{m_{5}}=0.$
(2.7)

Using Binet formula of $\\{x_{n}\\}_{n\geq 0}$, we have

$\displaystyle|A_{1}|\gamma^{m_{1}}$
$\displaystyle=|-A_{2}(\gamma^{m_{2}}-\delta^{m_{2}})-A_{3}(\gamma^{m_{3}}-\delta^{m_{3}})-A_{4}(\gamma^{m_{4}}-\delta^{m_{4}})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-A_{5}(\gamma^{m_{5}}-\delta^{m_{5}})-A_{1}\delta^{n_{1}}|<9X\gamma^{m_{2}},$

therefore

$\gamma^{m_{1}-m_{2}}<9X.$ (2.8)

Thus, since $m_{1}>m_{2}$, we get that $A<\gamma\leq\gamma^{m_{1}-m_{2}}<9X$.
Recalling that we might have to replace $X$ with $14X$, we get the final bound
in this case $A<126X$.

Case III If both $C_{5},C_{6}$ are non-zero. In this case we have

$C_{1}x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{4}x_{n_{4}}+C_{5}x_{n_{5}}+C_{6}x_{n_{6}}.$

If $n_{4}=n_{1}$, then
$(C_{1}-C_{4})x_{n_{1}}+C_{2}x_{n_{2}}+C_{3}x_{n_{3}}=C_{5}x_{n_{5}}+C_{6}x_{n_{6}}$.
If $n_{2}=0$ or $n_{3}=0$ or $n_{5}=0$ or $n_{6}=0$, then in all the
situations we will use the first case and get the bound $A\leq 28X$, otherwise
we replace
$(C_{1},C_{2},C_{3},C_{4},C_{5},C_{6})\longrightarrow(C_{1}-C_{4},C_{2},C_{3},C_{5},C_{6},0)$.
The only effect is that $28X$ is replaced with $56X$. If $n_{1}\neq n_{4}$,
then switch $C_{1}$ with $C_{4}$. if needed, we may assume that
$n_{1}=\max\\{n_{4},n_{1}\\}$, therefore $n_{1}>n_{4}$. We relabel our indices
$(n_{1},n_{2},n_{3},n_{4},n_{5},n_{6})$ as
$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})$ where $m_{1}>m_{2}>m_{3}\geq m_{4}\geq
m_{5}\geq m_{6}$, and the coefficients $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6}$
as $A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}$ and change signs to at most a couple
of them so that our equation is now

$A_{1}x_{m_{1}}+A_{2}x_{m_{2}}+A_{3}x_{m_{3}}+A_{4}x_{m_{4}}+A_{5}x_{m_{5}}+A_{6}x_{m_{6}}=0.$
(2.9)

Using Binet formula of $\\{x_{n}\\}_{n\geq 0}$, we have

$\displaystyle|A_{1}|\gamma^{m_{1}}$
$\displaystyle=|-A_{2}(\gamma^{m_{2}}-\delta^{m_{2}})-A_{3}(\gamma^{m_{3}}-\delta^{m_{3}})-A_{4}(\gamma^{m_{4}}-\delta^{m_{4}})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-A_{5}(\gamma^{m_{5}}-\delta^{m_{5}})-A_{6}(\gamma^{m_{6}}-\delta^{m_{6}})-A_{1}\delta^{n_{1}}|<11X\gamma^{m_{2}},$

therefore

$\gamma^{m_{1}-m_{2}}<11X.$ (2.10)

Thus, since $m_{1}>m_{2}$, we get that $A<\gamma\leq\gamma^{m_{1}-m_{2}}<11X$.
Recalling that we might have to replace $X$ with $28X$, we get the desired
conclusion which is $A<308X$. ∎

Proof of Theorem 1.2

From the above lemma we get $A$ is bounded. It is possible for small $A$ that
the equation has infinitely many solutions. But we will show that this is not
the case, it has finitely many solutions. Using the substitution
$(C_{1},C_{2},C_{3},C_{4},C_{5},C_{6})\longrightarrow(C_{1}-C_{4},C_{2},C_{3},C_{5},C_{6},0)$,
and relabelling some of the variables, we may assume that
$m_{1}>m_{2}>m_{3}\geq m_{4}\geq m_{5}\geq m_{6}$ and that equation (2.9)
holds. Then estimate (2.10) holds, So by using (2.10), we have

$m_{1}-m_{2}<\frac{\log(11X)}{\log\psi}.$ (2.11)

Using the Binet formula of balancing-like sequence in (2.9), we get

$\displaystyle\left\rvert\gamma^{m_{2}}\left(A_{1}\gamma^{m_{1}-m_{2}}+A_{2}\right)-\left(\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}\right)\right\lvert$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\left\rvert-
A_{3}(\gamma^{m_{3}}-\delta^{m_{3}})-A_{4}(\gamma^{m_{4}}-\delta^{m_{4}})-A_{5}(\gamma^{m_{5}}-\delta^{m_{5}})-A_{6}(\gamma^{m_{6}}-\delta^{m_{6}})\right\lvert$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq
8X\gamma^{m_{3}}.$ (2.12)

Since $m_{1}-m_{2}>0$, $A_{1}\gamma^{m_{1}-m_{2}}+A_{2}\neq 0$. Thus
$\left\rvert A_{1}\gamma^{m_{1}-m_{2}}+A_{2}\right\lvert\left\rvert
A_{1}\delta^{m_{1}-m_{2}}+A_{2}\right\lvert\geq 1.$ Since $\left\rvert
A_{1}\delta^{m_{1}-m_{2}}+A_{2}\right\lvert\leq 2X$, we get $\left\rvert
A_{1}\gamma^{m_{1}-m_{2}}+A_{2}\right\lvert\geq 1/2X$. Further

$\left\rvert\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}\right\lvert\leq\frac{2X}{\gamma^{m_{2}}}.$

Hence,

$\left\rvert\gamma^{m_{2}}\left(A_{1}\gamma^{m_{1}-m_{2}}+A_{2}\right)-\left(\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}\right)\right\lvert\geq\frac{\gamma^{m_{2}}}{2X}-\frac{2X}{\gamma^{m_{2}}}.$

Now, assume

$\frac{\gamma^{m_{2}}}{2X}-\frac{2X}{\gamma^{m_{2}}}\leq\frac{\gamma^{m_{2}}}{8X}.$
(2.13)

Then $\gamma^{2m_{2}}<6X^{2}$, so $\gamma^{m_{2}}<3X$. Hence

$m_{6}\leq m_{5}\leq m_{4}\leq m_{3}\leq m_{2}\leq\frac{\log(3X)}{\log\psi}.$
(2.14)

Using (2.11), we have

$m_{1}<\frac{\log(33X^{2})}{\log\psi}.$ (2.15)

If (2.13) not holds, then

$\frac{\gamma^{m_{2}}}{8X}\leq\frac{\gamma^{m_{2}}}{2X}-\frac{2X}{\gamma^{m_{2}}}\leq
8X\gamma^{m_{3}}.$

Then $\gamma^{m_{2}-m_{3}}\leq 64X^{2}$, so

$m_{2}-m_{3}\leq 2\frac{\log(8X)}{\log\psi}.$ (2.16)

Rewrite the (2.9), we have

$\displaystyle\left\rvert\gamma^{m_{3}}\left(A_{1}\gamma^{m_{1}-m_{3}}+A_{2}\gamma^{m_{2}-m_{3}}+A_{3}\right)-\left(\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{m_{3}}}\right)\right\lvert$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\left\rvert-
A_{4}(\gamma^{m_{4}}-\delta^{m_{4}})-A_{5}(\gamma^{m_{5}}-\delta^{m_{5}})-A_{6}(\gamma^{m_{6}}-\delta^{m_{6}})\right\lvert$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq
6X\gamma^{m_{4}}.$ (2.17)

Assume $A_{1}\gamma^{m_{1}-m_{3}}+A_{2}\gamma^{m_{2}-m_{3}}+A_{3}=0$. Let
$i=m_{2}-m_{3},j=m_{1}-m_{3}$. Then

$j\leq\frac{\log(704X^{3})}{\log\psi}~{}~{}~{}\text{and}~{}~{}~{}i\leq
2\frac{\log(8X)}{\log\psi}$

are bounded. Therefore one can find all polynomials
$A_{1}X^{j}+A_{2}X^{i}+A_{3}$ and took that polynomial which has a root
$\gamma$, where $\gamma$ is a quadratic unit of norm $1$. For these balancing-
like sequences, $\delta$ is also a root of the same polynomial so that the
left-hand side of (2) is zero for any $m_{3}$. This shows that also
$m_{4}=m_{5}=m_{6}=0$. Then we have

$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})=(m_{3}+j,m_{3}+i,m_{3},0,0,0)$

is a parametric family of solutions. From now on assume that
$A_{1}\gamma^{m_{1}-m_{3}}+A_{2}\gamma^{m_{2}-m_{3}}+A_{3}\neq 0$. Thus
$\left\rvert
A_{1}\gamma^{m_{1}-m_{3}}+A_{2}\gamma^{m_{2}-m_{3}}+A_{3}\right\lvert\left\rvert
A_{1}\delta^{m_{1}-m_{3}}+A_{2}\delta^{m_{1}-m_{3}}+A_{3}\right\lvert\geq 1.$
Since

$\left\rvert
A_{1}\delta^{m_{1}-m_{3}}+A_{2}\delta^{m_{1}-m_{3}}+A_{3}\right\lvert\leq 3X,$

which implies $\left\rvert
A_{1}\gamma^{m_{1}-m_{3}}+A_{2}\gamma^{m_{2}-m_{3}}+A_{3}\right\lvert\geq\frac{1}{3X}.$
Further,

$\left\rvert\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{m_{3}}}\right\lvert\leq\frac{3X}{\gamma^{m_{3}}}.$

Hence,

$\left\rvert\gamma^{m_{3}}\left(A_{1}\gamma^{m_{1}-m_{3}}+A_{2}\gamma^{m_{2}-m_{3}}+A_{3}\right)-\left(\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{m_{3}}}\right)\right\lvert\geq\frac{\gamma^{m_{3}}}{3X}-\frac{3X}{\gamma^{m_{3}}}.$

If
$\frac{\gamma^{m_{3}}}{3X}-\frac{3X}{\gamma^{m_{3}}}\leq\frac{\gamma^{m_{3}}}{6X}$,
then $\gamma^{2m_{3}}\leq 18X^{2}$, so $\gamma^{m_{3}}<5X$. Hence,

$m_{6}\leq m_{5}\leq m_{4}\leq m_{3}\leq\frac{\log(5X)}{\log\psi}.$ (2.18)

Using (2.11) and (2.16), we have

$m_{2}\leq\frac{\log(320X^{3})}{\log\psi}~{}~{}\text{and}~{}~{}m_{1}\leq\frac{\log(3520X^{4})}{\log\psi}.$
(2.19)

Now, assume
$\frac{\gamma^{m_{3}}}{6X}\leq\frac{\gamma^{m_{3}}}{3X}-\frac{3X}{\gamma^{m_{3}}}\leq
6X\gamma^{m_{4}}$, then $\gamma^{m_{3}-m_{4}}\leq 36X^{2}$. Hence,

$m_{3}-m_{4}\leq 2\frac{\log(6X)}{\log\psi}.$ (2.20)

Rewrite the (2.9), we have

$\displaystyle\left\rvert\gamma^{m_{4}}\left(A_{1}\gamma^{m_{1}-m_{4}}+A_{2}\gamma^{m_{2}-m_{4}}+A_{3}\gamma^{m_{3}-m_{4}}+A_{4}\right)-\left(\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{m_{3}}}+\frac{A_{4}}{\gamma^{m_{4}}}\right)\right\lvert$
$\displaystyle=\left\rvert-
A_{5}(\gamma^{m_{5}}-\delta^{m_{5}})-A_{6}(\gamma^{m_{6}}-\delta^{m_{6}})\right\lvert$
$\displaystyle\leq 4X\gamma^{m_{5}}.$ (2.21)

Assume
$A_{1}\gamma^{m_{1}-m_{4}}+A_{2}\gamma^{m_{2}-m_{4}}+A_{3}\gamma^{m_{3}-m_{4}}=0$.
Let $i=m_{3}-m_{4},j=m_{2}-m_{4},k=m_{1}-m_{4}$. Then

$k\leq\frac{\log(25,344X^{5})}{\log\psi},~{}~{}~{}~{}~{}j\leq\frac{\log(2304X^{4})}{\log\psi}~{}~{}~{}\text{and}~{}~{}~{}i\leq
2\frac{\log(6X)}{\log\psi}$

are bounded. Therefore one can find all polynomials
$A_{1}X^{k}+A_{2}X^{j}+A_{3}X^{i}+A_{4}$ and took that polynomial which has a
root $\gamma$, where $\gamma$ is a quadratic unit of norm $1$. For these
balancing-like sequences, $\delta$ is also a root of the same polynomial so
that the left-hand side of (2) is zero for any $m_{4}$. This shows that also
$m_{5}=m_{6}=0$. Then we have

$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})=(m_{4}+k,m_{4}+j,m_{4}+i,m_{4},0,0)$

is a parametric family of solutions. From now on assume that
$A_{1}\gamma^{m_{1}-m_{4}}+A_{2}\gamma^{m_{2}-m_{4}}+A_{3}\gamma^{m_{3}-m_{4}}+A_{4}\neq
0$. Thus

$\left\rvert
A_{1}\gamma^{m_{1}-m_{4}}+A_{2}\gamma^{m_{2}-m_{4}}+A_{3}\gamma^{m_{3}-m_{4}}+A_{4}\right\lvert\left\rvert
A_{1}\delta^{n_{m}-m_{4}}+A_{2}\delta^{m_{2}-m_{4}}+A_{3}\delta^{m_{3}-m_{4}}+A_{4}\right\lvert\geq
1.$

Since $\left\rvert
A_{1}\delta^{m_{1}-m_{4}}+A_{2}\delta^{m_{2}-m_{4}}+A_{3}\delta^{m_{3}-m_{4}}+A_{4}\right\lvert\leq
4X$, which implies

$\left\rvert
A_{1}\gamma^{m_{1}-m_{4}}+A_{2}\gamma^{m_{2}-m_{4}}+A_{3}\gamma^{m_{3}-m_{4}}+A_{4}\right\lvert\geq\frac{1}{4X}.$

Further,

$\left\rvert\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{m_{3}}}+\frac{A_{4}}{\gamma^{m_{4}}}\right\lvert\leq\frac{4X}{\gamma^{m_{4}}}.$

Hence,

$\left\rvert\gamma^{m_{4}}\left(A_{1}\gamma^{m_{1}-m_{4}}+A_{2}\gamma^{m_{2}-m_{4}}+A_{3}\gamma^{m_{3}-m_{4}}+A_{4}\right)-\left(\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{n_{3}}}+\frac{A_{4}}{\gamma^{n_{4}}}\right)\right\lvert\geq\frac{\gamma^{m_{4}}}{4X}-\frac{4X}{\gamma^{m_{4}}}.$

If
$\frac{\gamma^{m_{4}}}{4X}-\frac{4X}{\gamma^{m_{4}}}\leq\frac{\gamma^{m_{4}}}{8X}$,
then $\gamma^{2m_{4}}\leq 32X^{2}$, so $\gamma^{m_{4}}<6X$. Hence,

$m_{6}\leq m_{5}\leq m_{4}\leq\frac{\log(6X)}{\log\psi}.$ (2.22)

Using (2.11), (2.16) and (2.20), we have

$m_{3}\leq\frac{3\log(6X)}{\log\psi},~{}~{}m_{2}\leq\frac{\log(13824X^{5})}{\log\psi}~{}~{}\text{and}~{}~{}m_{1}\leq\frac{\log(152064X^{6})}{\log\psi}.$
(2.23)

Now, assume
$\frac{\gamma^{m_{4}}}{8X}\leq\frac{\gamma^{m_{4}}}{4X}-\frac{4X}{\gamma^{m_{4}}}\leq
4X\gamma^{m_{5}}$, then $\gamma^{m_{4}-m_{5}}\leq 32X^{2}$. Hence,

$m_{4}-m_{5}\leq\frac{\log(32X^{2})}{\log\psi}.$ (2.24)

Again, rewrite the equation (2.9), we have

$\displaystyle\Big{|}\gamma^{m_{5}}\left(A_{1}\gamma^{m_{1}-m_{5}}+A_{2}\gamma^{m_{2}-m_{5}}+A_{3}\gamma^{m_{3}-m_{5}}+A_{4}\gamma^{m_{4}-m_{5}}+A_{5}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{m_{3}}}+\frac{A_{4}}{\gamma^{m_{4}}}+\frac{A_{5}}{\gamma^{m_{5}}}\right)\Big{|}$
$\displaystyle=\left\rvert-A_{6}(\gamma^{m_{6}}-\delta^{m_{6}})\right\lvert$
$\displaystyle\leq 2X\gamma^{m_{6}}.$ (2.25)

Assume
$A_{1}\gamma^{m_{1}-m_{5}}+A_{2}\gamma^{m_{2}-m_{5}}+A_{3}\gamma^{m_{3}-m_{5}}+A_{4}\gamma^{m_{4}-m_{5}}+A_{5}=0$.
Let $i=m_{4}-m_{5},j=m_{3}-m_{5},k=m_{2}-m_{5},l=m_{1}-m_{5}$. Then

$l\leq\frac{\log(811008X^{7})}{\log\psi},~{}~{}~{}k\leq\frac{\log(73728X^{6})}{\log\psi},~{}~{}~{}j\leq\frac{\log(1152X^{4})}{\log\psi}~{}~{}~{}\text{and}~{}~{}~{}i\leq\frac{\log(32X^{2})}{\log\psi}$

are bounded. Therefore one can find all polynomials
$A_{1}X^{l}+A_{2}X^{k}+A_{3}X^{j}+A_{4}X^{i}+A_{5}$ and took that polynomial
which has a root $\gamma$, where $\gamma$ is a quadratic unit of norm $1$. For
these balancing-like sequences, $\delta$ is also a root of the same polynomial
so that the left-hand side of (2) is zero for any $m_{5}$. This shows that
also $m_{6}=0$. Then we have

$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})=(m_{5}+l,m_{5}+k,m_{5}+j,m_{5}+i,m_{5},0)$

is a parametric family of solutions. From now on assume that

$A_{1}\gamma^{m_{1}-m_{5}}+A_{2}\gamma^{m_{2}-m_{5}}+A_{3}\gamma^{m_{3}-m_{5}}+A_{4}\gamma^{m_{4}-m_{5}}+A_{5}\neq
0.$

Thus

$\displaystyle\left\rvert
A_{1}\gamma^{m_{1}-m_{5}}+A_{2}\gamma^{m_{2}-m_{5}}+A_{3}\gamma^{m_{3}-m_{5}}+A_{4}\gamma^{m_{4}-m_{5}}+A_{5}\right\lvert$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left\rvert
A_{1}\delta^{m_{1}-m_{5}}+A_{2}\delta^{m_{2}-m_{5}}+A_{3}\delta^{m_{3}-m_{5}}+A_{4}\delta^{m_{4}-m_{5}}+A_{5}\right\lvert\geq
1.$

Since $\left\rvert
A_{1}\delta^{m_{1}-m_{5}}+A_{2}\delta^{m_{2}-m_{5}}+A_{3}\delta^{m_{3}-m_{5}}+A_{4}\delta^{m_{4}-m_{5}}+A_{5}\right\lvert\leq
5X$, which implies

$\left\rvert
A_{1}\gamma^{m_{1}-m_{5}}+A_{2}\gamma^{m_{2}-m_{5}}+A_{3}\gamma^{m_{3}-m_{5}}+A_{4}\gamma^{m_{4}-m_{5}}+A_{5}\right\lvert\geq\frac{1}{5X}.$

Further,

$\left\rvert\frac{A_{1}}{\gamma^{m_{1}}}+\frac{A_{2}}{\gamma^{m_{2}}}+\frac{A_{3}}{\gamma^{m_{3}}}+\frac{A_{4}}{\gamma^{m_{4}}}+\frac{A_{5}}{\gamma^{m_{5}}}\right\lvert\leq\frac{5X}{\gamma^{m_{5}}}.$

Hence, the left-hand side of (2) is at least as large as

$\frac{\gamma^{m_{5}}}{5X}-\frac{5X}{\gamma^{m_{5}}}.$

If
$\frac{\gamma^{m_{5}}}{5X}-\frac{5X}{\gamma^{m_{5}}}\leq\frac{\gamma^{m_{5}}}{10X}$,
then $\gamma^{2m_{5}}\leq 50X^{2}$, so $\gamma^{m_{5}}<8X$. Hence,

$m_{6}\leq m_{5}\leq\frac{\log(8X)}{\log\psi}.$ (2.26)

Using (2.11), (2.16), (2.20) and (2.24), we have

$m_{4}\leq\frac{\log(256X^{3})}{\log\psi},~{}~{}m_{3}\leq\frac{\log(9216X^{5})}{\log\psi},~{}~{}m_{2}\leq\frac{\log(589824X^{7})}{\log\psi}~{}~{}\text{and}~{}~{}m_{1}\leq\frac{\log(6488064X^{8})}{\log\psi}.$
(2.27)

Now, assume
$\frac{\gamma^{m_{5}}}{10X}\leq\frac{\gamma^{m_{5}}}{5X}-\frac{5X}{\gamma^{m_{5}}}\leq
2X\gamma^{m_{6}}$, then $\gamma^{m_{5}-m_{6}}\leq 20X^{2}$. Hence,

$m_{5}-m_{6}\leq\frac{\log(20X^{2})}{\log\psi}.$ (2.28)

Again, rewrite the equation (2.9), we have

$\displaystyle\left\rvert\gamma^{m_{6}}\left(A_{1}\gamma^{m_{1}-m_{6}}+A_{2}\gamma^{m_{2}-m_{6}}+A_{3}\gamma^{m_{3}-m_{6}}+A_{4}\gamma^{m_{4}-m_{6}}+A_{5}\gamma^{m_{5}-m_{6}}+A_{6}\right)\right\rvert$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\left\rvert\delta^{m_{6}}\left(A_{1}\delta^{m_{1}-m_{6}}+A_{2}\delta^{m_{2}-m_{6}}+A_{3}\delta^{m_{3}-m_{6}}+A_{4}\delta^{m_{4}-m_{6}}+A_{5}\delta^{m_{5}-m_{6}}+A_{6}\right)\right\rvert$
(2.29)

Assume left-hand side of (2) is zero. Let
$i=m_{5}-m_{6},j=m_{4}-m_{6},k=m_{3}-m_{6},l=m_{2}-m_{6},m=m_{1}-m_{6}$. Then

$\displaystyle m$
$\displaystyle\leq\frac{\log(16220160X^{9})}{\log\psi},~{}~{}~{}l\leq\frac{\log(1474560X^{8})}{\log\psi},~{}~{}~{}k\leq\frac{\log(23040X^{6})}{\log\psi},$
$\displaystyle
j\leq\frac{\log(640X^{4})}{\log\psi}~{}~{}~{}\text{and}~{}~{}~{}i\leq\frac{\log(20X^{2})}{\log\psi}$

are bounded. Therefore one can find all polynomials
$A_{1}X^{m}+A_{2}X^{l}+A_{3}X^{k}+A_{4}X^{j}+A_{5}X^{i}+A_{6}$ and took that
polynomial which has a root $\gamma$, where $\gamma$ is a quadratic unit of
norm $1$. For such, (2) holds for all $m_{6}$. Hence, we have the parametric
family of solutions

$(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})=(m_{6}+m,m_{6}+l,m_{6}+k,m_{6}+j,m_{6}+i,m_{6}).$

From now on assume that left-hand side of (2) is non-zero. Thus

$\displaystyle\left\rvert
A_{1}\gamma^{m_{1}-m_{6}}+A_{2}\gamma^{m_{2}-m_{6}}+A_{3}\gamma^{m_{3}-m_{6}}+A_{4}\gamma^{m_{4}-m_{6}}+A_{5}\gamma^{m_{5}-m_{6}}+A_{6}\right\lvert$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left\rvert
A_{1}\delta^{m_{1}-m_{6}}+A_{2}\delta^{m_{2}-m_{6}}+A_{3}\delta^{m_{3}-m_{6}}+A_{4}\delta^{m_{4}-m_{6}}+A_{5}\delta^{m_{5}-m_{6}}+A_{6}\right\lvert\geq
1.$

Left–hand side second factor of the above equation is $\leq 6X$. Hence

$\left\rvert
A_{1}\gamma^{m_{1}-m_{6}}+A_{2}\gamma^{m_{2}-m_{6}}+A_{3}\gamma^{m_{3}-m_{6}}+A_{4}\gamma^{m_{4}-m_{6}}+A_{5}\gamma^{m_{5}-m_{6}}+A_{6}\right\lvert\geq\frac{1}{6X}.$

Hence, equation (2) becomes

$\frac{\gamma^{m_{6}}}{6X}\leq 6\delta^{m_{6}}=\frac{6X}{\gamma^{m_{6}}},$

which gives

$m_{6}\leq\frac{\log(6X)}{\log\psi}.$ (2.30)

Using (2.11), (2.16), (2.20), (2.24) and (2.28), we have

$\displaystyle m_{6}\leq\frac{\log(6X)}{\log\psi},$ $\displaystyle
m_{5}\leq\frac{\log(120X^{3})}{\log\psi},$ $\displaystyle
m_{4}\leq\frac{\log(3840X^{5})}{\log\psi},$ $\displaystyle
m_{3}\leq\frac{\log(138240X^{7})}{\log\psi},$ (2.31) $\displaystyle
m_{2}\leq\frac{\log(8847360X^{9})}{\log\psi},$ $\displaystyle
m_{1}\leq\frac{\log(97320960X^{10})}{\log\psi}.$

Note that (2) contains (2.26), (2.27), (2.22), (2.23), (2.18), (2.19), (2.14)
and (2.15). Recalling that we have to replace $X$ by $2X$, this completes the
proof of the Theorem (1.2).

##  3 Numerical example

Take $A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}\in\\{0,\pm 1\\}$. Hence $X=1$,
therefore $A\leq 308$. Thus, Theorem 1.2 says that the sporadic solutions are
of the form

$x_{m_{1}}\pm A_{2}x_{m_{2}}\pm A_{3}x_{m_{3}}\pm A_{4}x_{m_{4}}\pm
A_{5}x_{m_{5}}\pm A_{6}x_{m_{6}}=0,$

where $A_{2},A_{3},A_{4},A_{5},A_{6}\in\\{0,\pm 1\\}$, $m_{1}>m_{2}\geq
m_{3}\geq m_{4}\geq m_{5}\geq m_{6}\geq 0$. Here
$m_{6}<3,m_{5}<8,m_{4}<13,m_{3}<18,m_{2}<24$ and $m_{1}>m_{2}$. Now for
$r\in[3,308],m_{6}\in[0,3],m_{5}\in[m_{6},8],m_{4}\in[m_{5},13],m_{3}\in[m_{4},24],m_{2}\in[m_{3},24],\alpha_{1}\in\\{0,1\\}$
and $\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5}\in\\{0,\pm 1\\}$, we search
the equation

$x_{m_{1}}=\mid\alpha_{2}x_{m_{2}}+\alpha_{3}x_{m_{3}}+\alpha_{4}x_{m_{4}}+\alpha_{5}x_{m_{5}}+\alpha_{6}x_{m_{6}}\mid,$

holds for some $m_{1}>m_{2}$. Using Mathematica software and find only one
solution when $A=5$, that is,

$x_{2}=x_{1}+x_{1}+x_{1}+x_{1}+x_{1}.$

## References

  * [1] A. Behera and G. K. Panda, On the square roots of triangular numbers, _Fib. Quart._ 37 (1999), 98–105.
  * [2] M. Ddamulira, F. Luca, and R. Tichy, On the Pillai-Tijdeman Diophantine equation involving terms of Lucas sequences, arXiv:2105.01569v1.
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Department of Mathematics
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KIIT Deemed to be University, Bhubaneswar, India
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Department of Mathematics
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IISER Bhopal, Madhya Pradesh, India
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