# Generate then Retrieve: Conversational Response Retrieval Using LLMs as
Answer and Query Generators

Zahra Abbasiantaeb
University of Amsterdam
Amsterdam
The Netherlands
<EMAIL_ADDRESS>
&Mohammad Aliannejadi
University of Amsterdam
Amsterdam
The Netherlands
<EMAIL_ADDRESS>
Zahra Abbasiantaeb  Mohammad Aliannejadi
University of Amsterdam
Amsterdam
The Netherlands

###### Abstract

CIS is a prominent area in information retrieval (IR) which focuses on
developing interactive knowledge assistants. These systems must adeptly
comprehend the user’s information requirements within the conversational
context and retrieve the relevant information. To this aim, the existing
approaches model the user’s information needs with one query called rewritten
query and use this query for passage retrieval. In this paper, we propose
three different methods for generating multiple queries to enhance the
retrieval. In these methods, we leverage the capabilities of large language
models (LLMs) in understanding the user’s information need and generating an
appropriate response, to generate multiple queries. We implement and evaluate
the proposed models utilizing various LLMs including GPT-4 and Llama-2 chat in
zero-shot and few-shot settings. In addition, we propose a new benchmark for
TExt Retrieval Conference (TREC) Interactive Knowledge Assisstance Track
(iKAT) based on GPT-3.5 judgments. Our experiments reveal the effectiveness of
our proposed models on TREC iKAT dataset.

##  1 Introduction

Figure 1: An example conversation with a complex user request. The system in
this case generates three distinct queries from the initial answer and
searches for every query in the passage collection. Then it reads and reasons
over the top passages to generate the final grounded answer.

CIS is a well-established topic in information retrieval (IR) Zamani et al.
(2022), where a knowledge assistant interacts with the user to fulfill their
information needs. While conversations can be complex Radlinski and Craswell
(2017), involving various types of interactions such as revealment and
clarification, one of the main goals of the system is to provide accurate
responses to users’ queries during the conversation. The TExt Retrieval
Conference (TREC) Conversational Assistance Track (CAsT) 2019–2022 Dalton et
al. (2020) focuses on the development of conversational knowledge assistants,
while the main focus of this track is on evaluating passage retrieval and
dialogue context modeling. TREC Interactive Knowledge Assisstance Track (iKAT)
Aliannejadi et al. (2024) evolves CAsT into emphasizing the development of
personalized conversational knowledge assistants, where each dialogue is
coupled with a knowledge base describing the user. This knowledge base
contains various types of information, ranging from personal details to past
experiences and exchanges with the system Aliannejadi et al. (2024). It is
assumed that the user has revealed this information in their previous
interactions with the system, thereby impacting the system’s response to the
same request based on the user’s background.

To ensure the groundedness and accuracy of responses Semnani et al. (2023),
existing methods follow a  retrieve-then-generate (RG) pipeline, breaking the
task into several subtasks: dialogue context modeling, retrieval, and answer
generation Voskarides et al. (2019); Yu et al. (2021). The user’s information
need is often represented either by a single rewritten query Voskarides et al.
(2019); Yu et al. (2020) or by a single representation in the query embedding
space Yu et al. (2021); Hai et al. (2023).

Representing complex information needs using only one query leads to several
limitations, especially in cases where the information cannot be answered
using a single passage and requires complex reasoning over multiple facts in a
chain-of-thought scenario Aliannejadi et al. (2024); Lyu et al. (2023). Take
the user query of Figure 1 as an example. Clearly, it is unlikely to find a
passage that has distance information about all these universities compared to
the user’s address. Therefore, the system would need to gather relevant
information from different sources (i.e., issue multiple queries) and reason
over the gathered evidence to generate the final response. Existing RG methods
often fail to answer such complex queries Aliannejadi et al. (2024). This is
because existing ranking methods rely solely on semantic similarity between a
query and a passage, without high-level reasoning or control over the set of
retrieved passages. For example, they do not ensure that the top results
contain address information about all three universities the user is
interested in.

To address these challenges, in this work, we propose to rely on the knowledge
and capabilities of large language models to respond to complex user queries
in a dialogue. Specifically, inspired by the relevant literature Shuster et
al. (2022b), we study several methods based on the  generate-then-retrive (GR)
pipeline. We first prompt the LLM to generate an answer for a given dialogue,
followed by various retrieval strategies to ground the generated answer in the
passage collection. These strategies include: (i) using the generated answer
as a query to rank passages Gao et al. (2023); and (ii) a novel approach of
breaking the answer into several searchable queries. We hypothesize that
leveraging an LLM’s knowledge and reasoning capacity leads to a more accurate
response, enabling the model to generate more effective queries based on the
generated answer. For example, in Figure 1, the LLM’s answer that compares the
distance of the three universities helps it generate more effective queries,
as shown in the figure.

This leads us to our first research question: (RQ1) Can we leverage LLMs’
internal knowledge and reasoning capacity to enhance conversational passage
retrieval in a GR pipeline? To answer this question, we propose various GR
retrieval pipelines based on GPT-4 and LLaMA. Regarding the GR-based methods,
we address our next research question: (RQ2) Can we leverage LLMs’s reasoning
capabilities to generate related searchable queries to enhance retrieval? We
answer this question by comparing our proposed query generation method with
three alternatives: (i) using the generated answer as a query, (ii) directly
prompting the LLM to generate queries before generating an answer, and (iii)
prompting the LLM to generate the query rewrite. We conduct extensive
experiments on the iKAT 2023 dataset,111https://trecikat.com where we find
that GR pipelines significantly outperform their RG counterparts by a large
margin, and generating queries based on the LLM’s answer leads to more
effective queries and retrieval.

Given that only a handful of GR runs were evaluated in the official iKAT
assessment pool, we find that the released relevance assessments do not
accurately reflect the effectiveness of our proposed methods, leading to a
very high rate of unjudged passages in the top 10 results (72.78%). Recent
work shows that LLMs can accurately predict passage relevance Faggioli et al.
(2023); MacAvaney and Soldaini (2023), leading to high agreement with human
assessors. We follow the same approach and leverage GPT-3.5 in a one-shot
manner to assess all the passages in our new pool. To ensure our assessment is
not biased by the choice of the LLM, we conduct extensive experiments and
study the correlation and agreement with human labels (see appendix).

We summarize our contributions as follows:

  * •

We propose three GR-based conversational passage retrieval models, leveraging
the LLM’s internal knowledge to generate an answer and break it down into
multiple queries.

  * •

We implement our method using commercial and open-source LLMs, demonstrating
its effectiveness in both settings.

  * •

We investigate the usability of existing relevance judgments in evaluating GR-
based models. We collect and release GPT-3.5-based relevance judgments,
incorporating the new models into the judgment pool.222The code and data will
be released upon acceptance.

  * •

We conduct extensive experiments, showcasing the effectiveness of our proposed
approach under various conditions.

##  2 Related Work

Conversational information seeking. Recently, conversational information
seeking (CIS) has gained significant popularity in both IR and natural
language processing (NLP) communities Anand et al. (2020). Similar to
knowledge-intensive dialogues Dinan et al. (2019); Feng et al. (2021); Li et
al. (2022), a key challenge in CIS is to model the dialogue context to better
understand the user information need and perform effective retrieval Zamani et
al. (2022). As outlined by Radlinski and Craswell (2017), a CIS dialogue could
consist of numerous user–system interactions, such as revealment, information
request, and feedback, making it more complex for the existing systems to
model dialogue context and user information need. TREC CAsT 2019–2022 Dalton
et al. (2020) and iKAT 2023 Aliannejadi et al. (2024) aim at addressing these
challenges through a common evaluation framework where complex and knowledge-
intensive dialogues were provided to the participants, as well as several
passage collections. The goal was to retrieve relevant passages for each turn
in a dialogue and generate a response synthesizing several passages.

Retrieval pipeline. Most existing methods follow the two-step pipeline for
retrieval which includes first-stage retrieval and re-ranking. First-stage
retrieval aims to extract as many relevant passages from the collection. In
this step, the recall is more important, and models like BM25 are usually
used. The goal of re-ranking is to improve the precision by bringing the most
relevant passages to the top of the list of passages returned by retrieval.
The Cross-encoder model is one of the widely used re-ranker models.

Modeling dialogue context. Most existing methods tackle this problem by query
rewriting where the goal is to address the ambiguity and dependence of a user
utterance by resolving its dependencies and making it self-contained
Voskarides et al. (2019, 2020); Lin et al. (2021b). The rewritten query is
supposed to be a self-contained and context-independent query that represents
the user’s information needs per turn. CRDR Qian and Dou (2022) forms the
rewritten query by modifying the query by disambiguation of the anaphora and
ellipsis. The existing work trains GPT-2 Yu et al. (2020); Vakulenko et al.
(2021) and T5 Dalton et al. (2020) models on CANARD Elgohary et al. (2019) to
generate the rewritten query. The input of these models is created based on
different combinations of the previous user turns and system responses. The
CANARD dataset Elgohary et al. (2019) is widely used for training such models,
including the manually rewritten queries of QuAC Choi et al. (2018), which is
a conversational machine reading comprehension dataset that is collected
manually by pairing a teacher and a student annotator. Another line of
research aims at learning to represent the dialogue context directly for
passage retrieval Yu et al. (2021); Hai et al. (2023), where a distillation
loss learns to map the representation of the whole dialogue context to the one
of the gold resolved query, hence improving the dense retrieval performance.

Our work distinguishes itself from these works by leveraging the power of LLMs
to model dialogue context and uses the LLMs’ knowledge to answer the user’s
question directly first, and then try to ground its answer on the passage
collection.

Grounding LLM-generated responses. IR is a common approach to ground LLM-
generated responses and build grounded chatbots. BlenderBot 2–3 Shuster et al.
(2022b) utilizes the results of web search for this purpose, while SeeKer
Shuster et al. (2022a) employs a three-step approach in which it generates
search queries, extracts useful knowledge from the top passages, and generates
the final answer. Semnani et al. (2023) learns to avoid hallucination and
improve the factuality of the generated responses by searching and grounding
the responses on Wikipedia articles. Other approaches such as GenRead Yu et
al. (2023) study the effectiveness of LLMs in generating multiple passages for
a given question and use the generated passages to generate the final
response.

Differently, in this work, we focus on modeling the dialogue context through
LLMs’ response and query generation and use that to enhance passage retrieval
performance. Our goal is to leverage the LLM’s response as a means to model
and expand the user utterance, enhancing retrieval performance.

##  3 Methodology

Task definition. Each conversation revolves around a topic $t$ and starts with
a user utterance. A conversation includes several turns, where a turn starts
with a user utterance $u_{i}$, followed by a system response called $r_{i}$. A
conversation, which comprises a set of turns, is represented as
${(u_{1},r_{1}),...,(u_{n},r_{n})}$. The TREC iKAT dataset also contains the
user’s persona. The user’s persona is a knowledge base, consisting of a set of
statements shown as $PTKB=\\{s_{1},...,s_{l}\\}$, where each statement $s_{i}$
is a natural language sentence. The task of conversational assistants is
defined as follows: (i) retrieving relevant passages to the current user
utterance from the collection $D=\sum_{i=1}^{|D|}d_{i}$, and (ii) generating
the response $r_{i}$ given the user utterance $u_{i}$ and previous turns
${(u_{1},r_{1}),...,(u_{i-1},r_{i-1})}$, grounded on the retrieved passages.

Retrieve then generate. The raw user utterance cannot be used as a query for
the passage retrieval step, as it is not a self-contained query and depends on
the context Dalton et al. (2020). A common practice is to form the rewritten
query $q_{i}$ for the current user utterance $u_{i}$ Voskarides et al. (2019);
Yu et al. (2020). The rewritten query is used in a first-stage retrieval model
such as BM25 Robertson and Zaragoza (2009). The top passages returned by the
first-stage retriever are then passed to the re-ranker model. The top-k
passages returned by the re-ranker are finally passed to the response
generation model to generate the response $r_{i}$.

Generate then retrieve. Inspired by existing work Gao et al. (2023), we
propose a set of approaches that rely on the answer generated by the LLM. Our
goal is to leverage the LLM’s internal knowledge and reasoning capability to
improve passage retrieval, by generating a set of $n$ queries
$\\{q_{1}^{i},\dots,q_{n}^{i}\\}$ given the current user utterance $u_{i}$ and
the context of the conversation:

  * •

AD: In this approach, we treat the LLM’s answer $r_{i}^{\prime}$ as a single
long query and pass it to both the first-stage retrieval and re-ranker models.

  * •

QD: We prompt the LLM to directly generate a maximum of five queries to search
for the answer. Each query of turn $i$, denoted as $q^{i}_{j}$, is then passed
to both the first-stage retrieval model and the re-ranker model. The output of
the re-ranker for each query is demonstrated as
$d^{i,j}_{1},d^{i,j}_{2},\dots,d^{i,j}_{m}$. Subsequently, we interleave the
results of all the queries to obtain the final ranking:
$d^{i}_{1},d^{i}_{2},\dots,d^{i}_{k}$.

  * •

AQD: Here, we combine two other approaches: (i) prompting the LLM to generate
an initial response $r_{i}^{\prime}$, and (ii) prompting the LLM to generate
up to five queries to refine its own generated answer. Similar to the QD
model, we then pass these generated queries to the first-stage retrieval and
re-ranking model and interleave the ranked list of all queries.

  * •

AQD${}_{\text{A}}$: Interleaving the results of the generated queries is
suboptimal, as some of the generated queries may be of low quality. To tackle
this problem, we propose a variant to the AQD approach, where we re-rank the
final ranking list, based on the predicted relevance to the generated response
$r_{i}^{\prime}$.

Prompts: The GPT-4 model is used as a zero-shot learner for AQD and QD
approaches. The prompts used for these approaches are shown in Table 5 and 6,
respectively. The LLaMA model is given a few-shot prompt for query generation
in AQD and QD approaches as is shown in Table 9 and 10, respectively. The
examples of few-shots are from the pruned turns of the same dataset and output
of the GPT-4 model. For answer generation in AQD we design a zero-shot prompt.
The same answer generated in AQD approach is used for AD approach. Also, the
AQD and AQD${}_{\text{A}}$ approaches use the same prompt

##  4 Experimental Setup

Hyper-parameters. For the first-stage retrieval we employ the BM25 model from
Pyserini Lin et al. (2021a) using the default values for the parameters. For
the re-ranker, we use the pre-trained Cross-Encoder model ms-marco-
MiniLM-L-6-v2 from the sentence_transformers library with a maximum length of
512. We use LLaMA-chat 13B with the following parameters: top_k$=10$,
top_p$=0.9$, temperature$=0.75$. For the AQD approach, we design a two-shot
prompt for LLaMA. For AD and QR, we use LLaMA as a zero-shot learner and for
QD we design a one-shot in-context learning prompt for LLaMA. We use the GPT-4
model as a zero-shot learner using the default values of parameters for all
approaches. We utilize the GPT-3.5 with default parameters for prediction.

Dataset. We report the results on the TREC iKAT dataset, consisting of 25
conversations over 13 different topics and a total number of 133 turns. The
average length of conversations in iKAT is 13.04 which makes the context
modeling task more challenging. We evaluate our methods on iKAT as it is one
of the few datasets that features complex dialogues where single-query
rewriting is not effective.

Metric. We evaluate passage retrieval task using the official metrics of iKAT,
namely, nDCG@5, P@20, Recall@20, Recall, and mAP. nDCG@5 evaluates the
scenarios where the top passages are intended to be presented to the user,
while P@10 and Recall@20 measure the performance in cases where the top
results are going to be used by the LLM to generate an answer, assuming that
it would read and ground its answer on the top 20 passages.

GPT-3.5-based assessment. As mentioned earlier, the original pool of iKAT runs
contains only a handful of AQD models, resulting in a significant number of
unjudged passages among the top results of our proposed methods. For instance,
we observe that 82.40% of the top passages in AQD approach using LLaMA model
are unjudged (see Figure 2 for more details). Inspired by Faggioli et al.
(2023); MacAvaney and Soldaini (2023), we create a new pool of passages and
prompt GPT-3.5 to judge the relevance of passages. To make it a fair
comparison, we take the following considerations: (i) we judge all the
query–passage pairs even if they are already assessed in the original data,
and (ii) include the top 10 retrieved passages by all of our proposed models
and the baselines. The one-shot prompt used for relevance judgment is shown in
Table 8. In this prompt, we give the ground truth answer from the dataset as
an example of a passage with a score of 4 to the model. Overall, we judge the
relevance of 19,413 query–passage pairs.

Assessment bias. We argue that using GPT-3.5 as an assessor will not favor
GPT-4-based models Liu et al. (2023). We assess query–passage pairs that are
ranked by BM25 and the cross-encoder, while GPT-4 only generates the query in
such pipelines. Therefore, it is very unlikely that the judgments are favoring
any models, as the passages are not directly generated or ranked by GPT-4.

Baselines. Below we describe the baselines that were submitted as TREC runs,
as well as others that we implement in this work.

  * •

Human refers to using the resolved utterance provided in the gold dataset for
retrieval and re-ranking.

  * •

InfoSense refers to the TREC iKAT submission called
georgetown_infosense_ikat_run_3 Patwardhan and Yang (2023) and follows the GR
pipeline. In this run, relevant PTKB statements for each turn are determined.
We give the LLaMA-chat 13B model a one-shot prompt to generate an initial
answer given the context of the conversation and relevant PTKB statements. The
top passages are identified using the BM25 model.

  * •

LLaMA10 refers to the llama2_only_10_docs model Aliannejadi et al. (2024), a
baseline model proposed by the organizers of the track and is an RG model. It
prompts the LLaMA 7B model to generate the rewritten query. They use BM25 to
retrieve the top 1000 passages using this rewritten query. The top 10 passages
are re-ranked based on the judgment of the LLaMA model.

  * •

ConvGQR is another TREC baseline that uses a pre-trained model for query
rewriting and query expansion on the QReCC dataset Anantha et al. (2021).

  * •

monoT5 refers to TREC’s run_automatic_dense_monot5 model that uses a pre-
trained BART model for query rewriting. The BART model is fine-tuned on the
SAMSum Gliwa et al. (2019) and CANARD Elgohary et al. (2019) datasets. They
also employ a T5-based model for re-ranking.

  * •

AD baselines are also based on the GR pipeline. As described earlier, the LLM
is prompted to generate the answer given the context of the conversation and
the persona of the user. The generated response is passed to BM25 for first-
stage retrieval. The top 1000 passages returned by BM25 are re-ranked using
the pre-trained Cross-Encoder model.

  * •

QR baseline rewrite the query given the context of the conversation and
persona of the user. The GPT-4 and LLaMA-chat 13B models are used for this
approach. The rewritten query is then passed to the BM25 and Cross-Encoder
model to retrieve the relevant passages. The prompts designed for the zero-
shot QR model are shown in Table 7. This prompt is used for both GPT-4 and
LLaMA models.

Table 1: The results of passage retrieval task iKAT dataset on the GPT-3.5
benchmark for different pipelines (P.) and approaches (A.). [H]: using human-
rewritten queries, [G]: using answers generated by GPT-4. The superscripts
indicate the results of models that are significantly different, as determined
by a two-sided paired t-test with a Bonferroni correction, at a significance
level of $p<.05$. Given the space limit, we run the test only on the runs that
belong to the same block in the table (except for AD-GPT-4).

P. A. Model nDCG@5 nDCG P@20 Recall@20 Recall mAP GR QD (a) GPT-4
.5982${}^{\text{(cbd)}}$ .3839${}^{\text{(cbd)}}$ .6143${}^{\text{(b)}}$
.1259${}^{\text{(b)}}$ .3275${}^{\text{(d)}}$ .1942${}^{\text{(d)}}$ GR AQD
(b) GPT-4 .6300${}^{\text{(ad)}}$ .4020${}^{\text{(ad)}}$
.6414${}^{\text{(ad)}}$ .1313${}^{\text{(ad)}}$ .3384${}^{\text{(d)}}$ .2018
GR AQD (c) GPT-4 [H] .6594${}^{\text{(a)}}$ .4023${}^{\text{(ad)}}$ .6282
.1298 .3255${}^{\text{(d)}}$ .1964${}^{\text{(d)}}$ GR AQD${}_{\text{A}}$ (d)
GPT-4 .6826${}^{\text{(ab;n)}}$ .4799${}^{\text{(acb;n)}}$
.6124${}^{\text{(b;n)}}$ .1239${}^{\text{(b;n)}}$ .4771${}^{\text{(acb;n)}}$
.2079${}^{\text{(ac;n)}}$ GR QD (e) LLaMA .3925${}^{\text{(fghi)}}$
.2161${}^{\text{(fghi)}}$ .4184${}^{\text{(fghi)}}$ .0822${}^{\text{(fghi)}}$
.1947${}^{\text{(fghi)}}$ .0959${}^{\text{(fghi)}}$ GR AQD (f) LLaMA
.5033${}^{\text{(ei)}}$ .3004${}^{\text{(egi)}}$ .5421${}^{\text{(ei)}}$
.1092${}^{\text{(ei)}}$ .2657${}^{\text{(ei)}}$ .1458${}^{\text{(ei)}}$ GR AQD
(g) LLaMA [G] .5279${}^{\text{(e)}}$ .3184${}^{\text{(efi)}}$
.5331${}^{\text{(e)}}$ .1082${}^{\text{(e)}}$ .2743${}^{\text{(ei)}}$
.1503${}^{\text{(ei)}}$ GR AQD (h) LLaMA [H] .4851${}^{\text{(ei)}}$
.2974${}^{\text{(ei)}}$ .5124${}^{\text{(e)}}$ .1048${}^{\text{(e)}}$
.2562${}^{\text{(ei)}}$ .1380${}^{\text{(ei)}}$ GR AQD${}_{\text{A}}$ (i)
LLaMA .5571${}^{\text{(efh)}}$ .3906${}^{\text{(efgh)}}$
.5474${}^{\text{(ef)}}$ .1108${}^{\text{(ef)}}$ .4092${}^{\text{(efgh)}}$
.1771${}^{\text{(efgh)}}$ Baselines RG QR (j) Human .6193${}^{\text{(lm)}}$
.3422${}^{\text{(klm)}}$ .5658${}^{\text{(klm)}}$ .1173${}^{\text{(klm)}}$
.2877${}^{\text{(klm)}}$ .1593${}^{\text{(klm)}}$ RG QR (k) GPT-4
.5983${}^{\text{(lm)}}$ .2373${}^{\text{(jm)}}$ .5274${}^{\text{(jlm)}}$
.1071${}^{\text{(jlm)}}$ .1464${}^{\text{(jlm)}}$ .1164${}^{\text{(jm)}}$ RG
QR (l) LLaMA .4521${}^{\text{(jkm)}}$ .2431${}^{\text{(jm)}}$
.4538${}^{\text{(jkm)}}$ .0903${}^{\text{(jkm)}}$ .2211${}^{\text{(jkm)}}$
.1145${}^{\text{(jm)}}$ RG QR (m) LLaMA10 .3667${}^{\text{(jkl)}}$
.1081${}^{\text{(jkl)}}$ .3203${}^{\text{(jkl)}}$ .0621${}^{\text{(jkl)}}$
.0621${}^{\text{(jkl)}}$ .0512${}^{\text{(jkl)}}$ GR AD (n) GPT-4
.6433${}^{\text{(opqr;d)}}$ .4038${}^{\text{(opqr;d)}}$
.5793${}^{\text{(opqr;d)}}$ .1177${}^{\text{(opqr;d)}}$
.3739${}^{\text{(opqr;d)}}$ .1721${}^{\text{(opqr;d)}}$ GR AD (o) LLaMA
.5599${}^{\text{(nqr)}}$ .2504${}^{\text{(npq)}}$ .5342${}^{\text{(npqr)}}$
.1076${}^{\text{(npqr)}}$ .1705${}^{\text{(npqr)}}$ .1253${}^{\text{(npqr)}}$
GR AD (p) InfoSense .5274${}^{\text{(nqr)}}$ .2135${}^{\text{(nor)}}$
.4602${}^{\text{(noq)}}$ .0919${}^{\text{(noqr)}}$ .1407${}^{\text{(noqr)}}$
.1004${}^{\text{(noq)}}$ RG QR (q) ConvGQR .3978${}^{\text{(nopr)}}$
.2115${}^{\text{(nor)}}$ .3853${}^{\text{(nopr)}}$ .0686${}^{\text{(nopr)}}$
.2112${}^{\text{(nopr)}}$ .0835${}^{\text{(nopr)}}$ RG QR (r) monoT5
.4557${}^{\text{(nopq)}}$ .2676${}^{\text{(npq)}}$ .4316${}^{\text{(noq)}}$
.0770${}^{\text{(nopq)}}$ .2790${}^{\text{(nopq)}}$ .0986${}^{\text{(noq)}}$

##  5 Results and Discussions

Passage Ranking Results. The performance of the proposed models and the
baseline models are shown in Table 1. These results are based on the pool
generated by GPT-3.5 model (we provide the result of passage retrieval based
on the pool assessment of iKAT in Table 3). As can be seen, the AQD and
AQD${}_{\text{A}}$ models outperform the QR baseline using the LLaMA model
(addressing RQ1). In addition, the AQD${}_{\text{A}}$ approach outperforms the
QR approach using human query rewrites and the AD approach in terms of nDCG,
Recall, and mAP. In addition, the AQD${}_{\text{A}}$ model achieves a better
performance in terms of all metrics compared to the organizer’s baseline and
InfoSense (addressing RQ2).

According to Table 1, the LLaMA model in the AQD approach achieves a better
performance given the answers of GPT-4 model for query generation. This result
indicates the importance of the quality of the initially generated answer.
Given a wrong answer, the error will propagate to the query generation step
and reversely impact the quality of the generated queries.

Using the GPT-4 model as our LLM and the AQD${}_{\text{A}}$ approach, we have
achieved the best performance by outperforming the best baseline model based
on GR pipeline which is AD approach with GPT-4 as our LLM. (addressing RQ1,
RQ2) The AQD approach achieves a better performance in terms of mAP,
Recall@20, and P@20 compared to the AD approach. The results indicate that
using multiple queries for retrieval can help to retrieve more relevant
documents compared to using the initially generated answer as they cover
different aspects of the initially generated answer. By generating queries
from an answer, we are breaking the answer into several aspects, and by using
the output of the retrieval model for each of these queries, we are giving an
equal chance for these aspects to appear in the ranking list. The second stage
of re-ranking based on the initially generated answer is more effective than
interleaving the output of the re-ranker for each query. This makes sense
because the final goal here is to reconstruct the initial answer by grounding
it in the collection.

The superior performance of the AQD approach, compared to the QR approach
demonstrates the importance of having multiple queries rather than one query
for the complex user queries. The better performance of models with GR
pipeline compared to the models with RG pipeline represents the effectiveness
of using the knowledge and power of LLMs for the retrieval task.

The AQD model generates the queries in two steps, first doing reasoning and
generating the desired answer, and second generating queries from this answer.
In the QD approach the queries are directly generated from the context of the
conversation. The better performance of the AQD approach compared to the QD
approach indicates the importance of relying on the power of LLMs in context
modeling and reasoning for generating the answer. The answer generated by LLM
is a more helpful source for generating queries compared to the context of the
conversation.

The performance of the AQD approach using resolved utterance is presented in
Table 1 to study the power of LLMs in query understanding. Note that the
manual AQD-GPT-4 model performs slightly better than the automatic AQD-GPT-4
model in terms of lower cutoffs of Precision, Recall, and nDCG. While the
automatic AQD-GPT-4 model achieves a better performance in terms of Recall and
mAP. The result of all the metrics for the manual run of the LLaMA model is
lower than the automatic run where the LLaMA model relies on its own knowledge
to understand the context of the conversation and generate an answer.

Performance of different QR models shows that the LLaMA and GPT-4 models as
zero-shot learners cannot outperform humans in query rewriting tasks.

Pooling. To demonstrate the extent of lacking relevance judgments in the runs,
here we report the Judged@10 of some of the runs. The results of all the runs
are available in Table 3 in the appendix. Judged@10 simply reports the average
number of judged passages (between 0 and 10) in the top 10 passages of each
run. We see in Figure 2 that the number of judged passages in the top 10
passages returned by our proposed models is very low ($\sim$2), compared to
the models that participated at iKAT (AQD-GPT-4 and InfoSense), leading to
unfair evaluation. This is evident in Table 3 where we report the performance
of all the models using the official iKAT relevance labels, where we see that
all our proposed approaches are inferior to the iKAT runs, which is expected
given the very low rate of judged passages. Therefore, we generated a new
assessment pool using GPT-3.5 model, containing a total number of 19,413
relevance judgments over 133 conversational turns. To demonstrate the extent
to which the GPT-3.5-judgments agree with human labels, we re-generate the
pool of TREC iKAT benchmark and compare the performance of GPT-3.5 one-shot
model as an assessor with humans. According to Table 2, the GPT-3.5 and human
assessors have 66.84% agreement in terms of binary relevance judgment.
Additionally, the GPT-3.5 tends to give a higher relevance score compared to
humans.

Analysis. Performance of the proposed models per depth of the conversation and
topic are shown in Figures 4 and 3. As can be seen, as the conversation
continues and the context modeling becomes more challenging (turn$>$14), the
AQD and AQD${}_{\text{A}}$ approaches represent a better retrieval performance
compared to the AD model.

Table 2:  Confusion matrix comparing binary relevance judgments made by TREC
assessors and one-shot GPT-3.5 on TREC iKAT 2023.

|  | TREC iKAT 2023
---|---|---
|  | Relevant | Irrelevant
GPT-3.5 | Relevant | 12,885 |  0450
Irrelevant |  07146 | 2,203

Figure 2: Judged@10 metric in TREC iKAT official relevance assessments for the
first 10 passages returned by each model. Figure 3: Performance of the
proposed models per each conversation ((G): GPT-4). Figure 4: Performance of
the proposed models per each conversational depth ((G): GPT-4)

##  6 Conclusion

We propose two different models based on the GR pipeline for enhancing the
retrieval performance of CIS. The proposed models work by generating multiple
queries rather than using one rewritten query. We rely on the power of LLMs to
understand the user’s information needs and generate an appropriate response.
In AQD approach, we prompt LLM to answer the user’s question and then generate
multiple to refine that answer. We do retrieval and re-ranking for all queries
and interleave the ranking output for all queries. In AQD${}_{\text{A}}$
approach, which is a variant of AQD model, after retrieving the relevant
passages for each query, we re-rank the top passages for these queries using
the initial answer. In QD approach we directly prompt the LLM to generate the
queries given the context of the conversation and the user utterance. We
conduct extensive experiments to study the performance of the proposed
approaches. The experiments demonstrate the effectiveness of generating
multiple queries compared to one rewritten query. Additionally, we propose a
new assessment pool for iKAT dataset generated by GPT-3.5.

##  7 Limitations

We present two Generate-then-Retrieve-based methods for improving the
retrieval performance. Our proposed methods rely on the answer generated by
LLM and try to ground the generated answer to the given collection in
different ways. So, the retrieval is biased toward the answer generated by
LLM, and any error or limitation in the answer generation would influence the
retrieval and re-ranking. For example, if the LLM does not know about a
specific topic, it cannot generate a correct and complete answer. The low-
quality answer will result in generating low-quality and non-relevant queries
which decreases the performance of retrieval. Our main focus in this work is
on improving the performance of retrieval and we have not studied the quality
of the answer generation using our proposed models for generating the final
answer. In addition, we prompt the LLM to generate a maximum number of 5
queries. The impact of the number of queries is not studied in this work. The
impact of different numbers of queries on the performance of retrieval and
generating the final answer to the user question remains an interesting future
work.

##  8 Ethical considerations

Stressing the need to study and measure biases in Language Models (LLMs) when
generating data, we think it could cause unexpected ethical issues.
Consequently, we need to study the potential biases that exist in the data and
formalize their impact on the final output of the model. While in this study
we propose to use the answers and queries generated by LLMs for retrieval
models, we think these methods should be used carefully in real-world
retrieval systems, and designers should consider these biases.

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##  Appendix A Appendix

###  A.1 iKAT Benchmark

The retrieval performance of the proposed models using the benchmark of iKAT
is presented at Table 3. The percentage of unjudged passages among the first
10 retrieved passages for each model is written in this table.

Table 3: The results of passage retrieval task for iKAT dataset using the iKAT
benchmark. [H]: using human-rewritten queries, [G]: using answers generated by
GPT-4.

P. A. Model nDCG@5 nDCG P@20 Recall@20 Recall mAP Judged@10 GR QD GPT-4 .2122
.2572 .1820 .1041 .3176 .0954 3.42 GR AQD GPT-4 .4213 .3469 .3327 .1778 .3584
.1706 10.00 GR AQD GPT-4 [H] .2175 .2633 .1951 .1199 .3304 .0990 3.22 GR
AQD${}_{\text{A}}$ GPT-4 .2023 .3082 .1553 .0900 .4914 .0920 2.77 GR QD LLaMA
.0492 .1002 .0571 .0294 .1525 .0220 1.45 GR AQD LLaMA .1089 .1434 .0966 .0542
.1893 .0411 1.76 GR AQD LLaMA [G] .1432 .1912 .1312 .0709 .2546 .0616 2.69 GR
AQD LLaMA [H] .1151 .1707 .1147 .0681 .2293 .0514 2.06 GR AQD${}_{\text{A}}$
LLaMA .1149 .1970 .1162 .0609 .3196 .0575 1.97 Baselines RG QR Human .3010
.3538 .2726 .1590 .4979 .1476 5.05 RG QR GPT-4 .2124 .1853 .1921 .1005 .1863
.0828 3.79 RG QR LLaMA .1176 .1580 .1056 .0508 .2441 .0506 2.31 RG QR LLaMA10
.1466 .0756 .1192 .0553 .0553 .0376 10.00 RG QR ConvGQR .1623 .1518 .1421
.0611 .2034 .0551 10.00 RG QR monoT5 .2206 .2147 .1831 .0812 .3058 .0754 10.00
GR AD GPT-4 .1441 .2420 .1244 .0771 .4041 .0699 2.28 GR AD LLaMA .1304 .1479
.1395 .0738 .1675 .0577 2.53 GR AD InfoSense .3109 .2097 .2519 .1168 .1862
.1042 10.00

###  A.2 GPT-3.5 Pool

The comparison between the graded relevance scores given by TREC assessors and
GPT-3.5 is provided in Table 2.

Table 4:  Confusion matrix comparing graded relevance judgments made by TREC assessors and one-shot GPT-3.5 on TREC iKAT 2023.  |  | TREC iKAT 2023
---|---|---
|  | 0 | 1 | 2 | 3 | 4
GPT-3.5 | 0 | 8,743 | 227 | 112 |  039 |  09
1 | 3,515 | 400 | 209 |  070 | 11
2 | 3,189 | 668 | 450 | 189 | 47
3 | 1,625 | 630 | 476 | 284 | 52
4 |  0,620 | 414 | 369 | 284 | 52

###  A.3 Prompts

The prompt used for AQD and AD approach using GPT-4 is shown in Table 5. We
use the same prompt for AQD${}_{\text{A}}$ approach. The prompt used for zero-
shot QD using GPT-4 model is shown in Table 6. The term $ctx$ in the prompts
designed for GPT-4 includes all of the previous user utterances and system
responses.

For QR model the prompt shown in Table 7 is designed. This prompt is used for
both LLaMA and GPT-4 models. We pass all the previous user and system
interactions as $ctx$ in this prompt. The prompt designed for relevance
judgement using GPT-3.5 model is shown in Table 8. In this prompt the resolved
utterance, relevant PTKB statements, and the canonical answer from gold data
are used.

The two-shot prompt designed for LLaMA AQD approach is shown in Table 9. In
this prompt the answer generated by LLaMA itself is passed as the $response$.
For the prompts of LLaMA model, all the previous user utterances with the last
system response are passed as context to the model.

The one-shot prompt designed for LLaMA QD approach is shown in Table 10.

Table 5: The prompt designed for AQD and AD approaches using GPT-4 as zero-
shot learner. (1) Initial Answer Generation and (2) Query Generation in AQD
approach.
---
(1) # Instruction:
I will give you a conversation between a user and a system. Also, I will give
you some background information about the user. You should answer the last
question of the user. Please remember that your answer to the last question of
the user shouldn’t be more than 200 words.
# Background knowledge: {$ptkb$}
# Context: {$ctx$}
# User question: {$user\ utterance$}
# Response:
(2) # Can you generate the unique queries that can be used for retrieving your
previous answer to the user? (Please write each query in one line and don’t
generate more than 5 queries)
# Generated queries:
Table 6: The prompt designed for QD approach using GPT-4 as zero-shot learner.
Query Generation in QD approach.
---
# Instruction:
I will give you a conversation between a user and a system and some background
information about the user. Imagine you want to find the answer to last user
question by searching the google. You should generate the search queries that
you need to search in google. Please don’t generate more than 5 queries and
write each query in one line.
# Background knowledge: {$ptkb$}
# Context: {$ctx$}
# User question: {$user\ utterance$}
# Generated queries:
Table 7: The prompt designed for QR using GPT-4 and LLaMA models as zero-shot
learner. Query re-writing (QR).
---
# Instruction:
I will give you a conversation between a user and a system. Also, I will give
you some background information about the user. You should rewrite the last
question of the user into a self-contained query.
# Background knowledge: {$ptkb$}
# Context: {$ctx$}
# Please rewrite the following user question: {$user\ utterance$}
# Re-written query:
Table 8: The prompt designed for relevance judgement using GPT-3.5 as one-shot
learner. Relevance judgement.
---
# Instruction:
I will give you a user question and a passage, you should say to what extent
the given passage is relevant for answering the question by giving an integer
rate between 0-4. The relevance score lower than 2 means the document is not
relevant. I will also give some background information about the user who
asked this question.
# Background knowledge:
{$relevant\ ptkb\ statements$}
# User question: {$resolved\ utterance$}
# Document 1: {$canonical\ response$}
# Score: 4
# Document 2: {$document$}
# Score:
Table 9: The prompt designed for two-shot AQD using LLaMA model. The two-shot
prompt for query generation in AQD.
---
# Instruction:
Generate the unique queries to search them in a search engine to retrieving
the last response of the system to the user. (Please write each query in one
line and don’t generate more than 5 queries)
# Example 1
# Background knowledge: 1: My sister is following the ‘West Worl’, but I don’t
like it, 2: Johnny Depp made the Pirates of the Caribbean excellent, 3: My
friend suggested to me the ‘Now you see me’ movie, …
# Context:
user: Can you tell me what the Golden Globe Awards is?
user: What is it?
user: Is it different from the Oscars?
user: What is the difference between them?
user: No, I mean Academy Awards and Golden Globe Awards.
user: What else?
system: The Hollywood Foreign Press Association, a group of 93 journalists
from around 55 countries, ….
# User question: Did any of my favorite actresses win any of them?
# System response: Yes, both Jennifer Aniston and Lisa Kudrow, who you enjoyed
in the Friends series, have won Golden Globe Awards. Jennifer Aniston won the
Golden Globe in 2003 for Best Actress in a Television Series – Musical or
Comedy for Friends. Lisa Kudrow also won this category in 1998 for the same
series. Aniston’s performance in The Morning Show also earned her another
nomination in 2020. On the other hand, neither of the actresses have won an
Academy Award as of yet.
# Generated queries:
1\. Has Jennifer Aniston ever won a Golden Globe or an Academy Award?
2\. Has Lisa Kudrow ever won a Golden Globe or an Academy Award?
3\. List of Golden Globe winners in Best Actress in a Television Series –
Musical or Comedy category for 1998 and 2003.
4\. Did Jennifer Aniston win any awards for ‘The Morning Show’?
# Example 2
# Background knowledge: 1: I am diagnosed with diabetes type 2, 2: My husband
is a light drinker, 3: We have pasta twice a week! It is my favorite dish,….
# Context:
user: Can you tell me about different types of alcoholic drinks?
user: What is the difference between them?
user: No, the other category.
user: Ok. Can you compare base liquors and the third one?
user: Interesting, but I was mainly looking for ingredients and flavor! FYI, I
don’t drink myself.
user: How is it different from Liqueurs?
user: Why are their names so similar?
user: How about the percentage of alcohol?
system: Liquor is hard (the hardest) alcohol product made by distillation,
often clocking ….
# User question: Which types are suitable for my husband?
# System response: Considering your husband is a light drinker, he might enjoy
lower-alcohol content beverages such as wine, beer, or certain liqueurs.
Specifically, since he enjoys seafood like salmon, white wine such as
chardonnay, pinot grigio, or a light-bodied beer might be a good match.
However, always remember to consume alcohol in moderation, and if there are
any health concerns, consult with a physician.
# Generated queries:
1\. Which alcoholic beverages are suitable for a light drinker who likes
salmon?
2\. What types of alcohol go well with seafood?
3\. What types of alcohol are preferred for people who drink lightly?
4\. What alcoholic drinks have lower alcohol content?
5\. Recommendations for alcoholic beverages for light drinkers.
# Example 3
# Background knowledge: {$ptkb$}
# Context: {$ctx$}
# User question: {$user\ utterance$}
# System response: {$response$}
# Generated queries:
Table 10: The prompt designed for one-shot QD using LLaMA model. The one-shot
prompt designed for QD.
---
# Instruction:
Please generate self-contained questions that should be searched in a search
engine to answer the user’s LAST utterance. (Please write each query in one
line and don’t generate more than 5 queries)
# Example 1
# Background knowledge: 1: My sister is following the ‘West Worl’, but I don’t
like it, 2: Johnny Depp made the Pirates of the Caribbean excellent, 3: My
friend suggested to me the ‘Now you see me’ movie, it was fantastic, 4: I went
on a biking trip last year, 5: I usually like to drink coffee in the morning,
6: I watched the proposal and enjoyed it. Ryan Reynolds is my favorite!, 7:
The ‘Friends’ series was terrific, Jennifer Aniston and Lisa Kudrow were the
best stars!
# Context:
user: Can you tell me what the Golden Globe Awards is?
user: What is it?
user: Is it different from the Oscars?
user: What is the difference between them?
user: No, I mean Academy Awards and Golden Globe Awards.
user: What else?
system: The Hollywood Foreign Press Association, a group of 93 journalists
from around 55 countries, are the committee for the Globes. On the contrary,
the voting body of the Academy Awards; the Academy of Motion Picture Arts and
Sciences (AMPAS) consists of 6,000 voting members. The Golden Globes Award
recognizes the excellence of artists in both the film and television industry
in the United States as well as in other countries. However, Academy Awards
only recognize the excellence of artists in their cinematic achievements,
primarily in Hollywood or the American film industry. There are 25 categories
for Golden Globes; 14 in motion pictures and 11 in television. At present, the
Academy Awards has 24 categories. Usually, the Golden Globes ceremony is held
in January of each year while the Academy Awards ceremony is held in February
of each year.
# User question: Did any of my favorite actresses win any of them?
# Generated queries:
1\. Has Jennifer Aniston ever won a Golden Globe or an Academy Award?
2\. Has Lisa Kudrow ever won a Golden Globe or an Academy Award?
3\. Did Jennifer Aniston win any awards for ’The Morning Show’?
4\. List of Golden Globe winners in Best Actress in a Television Series –
Musical or Comedy category for 1998 and 2003.
# Example 2
# Background knowledge: {$ptkb$}
# Context: {$ctx$}
# User question: {$user\ utterance$}
# Generated queries:

  *[IR]: information retrieval
  *[TREC]: TExt Retrieval Conference
  *[iKAT]: Interactive Knowledge Assisstance Track
  *[CAsT]: Conversational Assistance Track
  *[RG]: retrieve-then-generate
  *[GR]: generate-then-retrive
  *[LLM]: large language model
  *[LLMs]: large language model
  *[CIS]: conversational information seeking
  *[NLP]: natural language processing


About special elements in quaternion algebras
over finite fields

Diana SAVIN

Abstract. In this paper we study special Fibonacci quaternions and special
generalized Fibonacci-Lucas quaternions in quaternion algebras over finite
fields.

Key Words: quaternion algebras; finite fields; Fibonacci numbers; Lucas
numbers, Fibonacci quaternions, generalized Fibonacci-Lucas quaternions.

2000 AMS Subject Classification: 15A06, 16G30, 1R52, 11B39.

1\. Introduction

Let $K$ be an algebraic number field and let
$\mathbb{H}_{K}\left(\alpha,\beta\right)~{}$be the generalized quaternion
algebra over the field $K.$ An arbitrary element from this algebra has the
form $x=x_{1}\cdot 1+x_{2}e_{2}+x_{3}e_{3}+x_{4}e_{4},$ where $x_{i}\in
K,i\in\\{1,2,3,4\\}$, where the elements of the basis
$\\{1,e_{2},e_{3},e_{4}\\}$ satisfy the following multiplication table:

$\cdot\,\,$ | $1$ | $e_{2}$ | $e_{3}$ | $e_{4}$
---|---|---|---|---
$1$ | $1$ | $e_{2}$ | $e_{3}$ | $e_{4}$
$e_{2}$ | $e_{2}$ | $\alpha$ | $e_{4}$ | $\alpha e_{3}$
$e_{3}$ | $e_{3}$ | $-e_{4}$ | $\beta$ | $-\beta e_{2}$
$e_{4}$ | $e_{4}$ | $-\alpha e_{3}$ | $\beta e_{2}$ | $-\alpha\beta$

Let $x$$\in$$\mathbb{H}_{K}\left(\alpha,\beta\right)$ and let
$\boldsymbol{n}\left(x\right)$ be the norm of a generalized quaternion $x.$
This norm has the following expression
$\boldsymbol{n}\left(x\right)=x_{1}^{2}-\alpha x_{2}^{2}-\beta
x_{3}^{2}+\alpha\beta x_{4}^{2}.$We recall that
$\mathbb{H}_{K}\left(\alpha,\beta\right)$ is a division algebra if and only if
for $x\in\mathbb{H}_{K}\left(\alpha,\beta\right)$ we have
$\boldsymbol{n}\left(x\right)=0$ if and only if $x=0.\,$ Otherwise, the
algebra $\mathbb{H}_{K}\left(\alpha,\beta\right)$ is called a split algebra.
When $\alpha=\beta=-1$ and $K=\mathbb{R}$ we obtain Hamilton quaternion
algebra $\mathbb{H}_{\mathbb{R}}\left(-1,-1\right),$ with the basis
$\\{1,i,j,k\\}.$ It is known that this algebra is a division algebra.
When $\alpha=\beta=-1$ and $K$ is a finite field ($K=\mathbb{Z}_{p},$ where
$p$ is a prime positive integer) we obtain the quaternion algebra
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$ It is known that this algebra
is a split algebra ([Gr, Mi, Ma; 15], Theorem 2, p. 882).

Let $\left(f_{n}\right)_{n\geq 0}$ be the Fibonacci sequence:

$f_{0}=0;f_{1}=1;f_{n}=f_{n-1}+f_{n-2},\;n\geq 2$

and $\left(l_{n}\right)_{n\geq 0}$ be the Lucas sequence:

$l_{0}=2;l_{1}=1;l_{n}=l_{n-1}+l_{n-2},\;n\geq 2$

In his paper [Ho; 63], A. F. Horadam introduced the Fibonacci quaternions
$\left(F_{n}\right)_{n\geq 0}:$

$F_{n}=f_{n}+f_{n+1}\cdot i+f_{n+2}\cdot j+f_{n+3}\cdot k,\;n\geq 0.$

Some properties of Fibonacci quaternions in certain quaternion algebras were
investigated in the papers [Ak, Ko, To; 14], [Ta; Ya; 15], [Po, Ki, Ke; 16],
[Y$\ddot{u}$, Ay; 16], etc. In this paper we determine the Fibonacci
quaternions (respectively generalized Fibonacci- Lucas quaternions) which are
zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ and the Fibonacci quaternions
which are invertible in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$
We recall some necessary results about Fibonacci sequence and about Fibonacci
quaternions.

Proposition 1.1 ([Fib]). Let $\left(f_{n}\right)_{n\geq 0}$ be the Fibonacci
sequence. If $f_{n}$ is a prime number ($n\geq 5$), then $n$ is a prime
number.

Proposition 1.2 ([Fib]). Let $\left(f_{n}\right)_{n\geq 0}$ be the Fibonacci
sequence. If $n,m$$\in$$\mathbb{N}^{*},$ $n|m,$ then $f_{n}|f_{m}.$

Theorem 1.1 ([Su], [Su; 92]). Let $p\notin\left\\{2,5\right\\}$ be a prime
positive integer. Then

$f_{\frac{p-\left(\frac{p}{5}\right)}{2}}\equiv\left\\{\begin{array}[]{ll}\
0\>(mod\>p),if\>p\equiv 1(mod4),\\\ \
2\left(-1\right)^{\left[\frac{p+5}{10}\right]}\cdot\left(\frac{p}{5}\right)\cdot
5^{\frac{p-3}{4}}\>(mod\>p),$ if $\>p\equiv 3(mod4)\end{array}\right.$

where $\left[x\right]$ is the integer part of x and $\left(\frac{p}{5}\right)$
is the Legendre symbol.

Proposition 1.3 ([Fib]). The cycle of the Fibonacci numbers mod $2$ is

$0,1,1(0,1,1),...$

so:
i) the cycle-length of the Fibonacci numbers mod $2$ is $3;$
ii) a Fibonacci number $f_{n}$ is even if and only if $n$$\equiv$$0$ (mod
$3$).
iii) a Lucas number $l_{n}$ is even if and only if $n$$\equiv$$0$ (mod $3$).

Proposition 1.4 ([Fib]). The cycle of the Fibonacci numbers mod $3$ is

$0,1,1,2,0,2,2,1(0,1,1,2,0,2,2,1),...$

so, the cycle-length of the Fibonacci numbers mod $3$ is $8.$

Proposition 1.5 ([Fib]). The cycle of the Fibonacci numbers mod $5$ is

$0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1(0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1),...$

so:
i) the cycle-length of the Fibonacci numbers mod $5$ is $20;$
ii) $f_{n}$$\equiv$$0$ (mod $5$) if and only if $n$$\equiv$$0$ (mod $5$).

Proposition 1.6 ([Ho; 63]). Let $\left(f_{n}\right)_{n\geq 0}$ be the
Fibonacci sequence and $F_{n}(n\geq 0)$ be the Fibonacci quaternions. Then,
the norm of the nth Fibonacci quaternions is:

$n\left(F_{n}\right)=3f_{2n+3}.$

Let $p$ be an odd prime positive integer. In the paper [Mi, Se; 11], C. J.
Miguel, R. Serodio calculated the number of zero-divisors in
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$ They proved that:

Theorem 1.2 ([Mi, Se; 11]). The number of zero-divisors in
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ is:

$p^{3}+p^{2}-p.$

Let $p$ be a prime positive integer. According to [Da, Dr; 70], if $f_{z}$ is
the smallest non-zero Fibonacci number with the property $p|f_{z},$ then
$z=z\left(p\right)$ is defined as the entry point of $p$ in the Fibonacci
sequence.

Remark 1.1. ([Da, Dr; 70]). $p|f_{n}$ if and only if $z\left(p\right)|n,$
$\left(\forall\right)n\in\mathbb{N}.$

Definition 1.1. ([Da, Dr; 70]). Let $\left(f_{n}\right)_{n\geq 0}$ be the
Fibonacci sequence. If we reduce the Fibonacci sequence modulo $p,$ to obtain
a periodic sequence. The period $k=k\left(p\right)$ is the smallest integer
$k$ for which

$f_{k}\equiv 0\ \left(mod\ p\right)\ and\ f_{k+1}\equiv 1\ \left(mod\
p\right).$

For example: $k\left(2\right)=z\left(2\right)=3,$ $k\left(3\right)=8,$
$z\left(3\right)=4,$ $k\left(5\right)=20,$ $z\left(5\right)=5$ (according to
Proposition 1.3, Proposition 1.4, Proposition 1.5).

Remark 1.2. ([Da, Dr; 70]). The entry point $z\left(p\right)$ divides the
period $k\left(p\right).$

Remark 1.3. ([Da, Dr; 70]). Let $p$ be an odd prime positive integer. The
following statements are true:
i) $k\left(p\right)=z\left(p\right)$ if $z\left(p\right)\equiv 2$ (mod $4$);
ii) $k\left(p\right)=2z\left(p\right)$ if $z\left(p\right)\equiv 0$ (mod $4$);
iii) i) $k\left(p\right)=4z\left(p\right)$ if $z\left(p\right)$ is odd.

Proposition 1.7. ([Fib]).Let $(f_{n})_{n\geq 0}$ be the Fibonacci sequence and
$(l_{n})_{n\geq 0}$ be the Lucas sequence Then:
i)

$f^{2}_{n}+f^{2}_{n+1}=f_{2n+1},\forall~{}n\in\mathbb{N};$

ii)

$f_{n}+f_{n+4}=3f_{n+2},\forall~{}n\in\mathbb{N};$

iii)

$l^{2}_{n}+l^{2}_{n+1}=5f_{2n+1},\forall~{}n\in\mathbb{N};$

iv)

$f_{m}l_{m+p}=f_{2m+p}+\left(-1\right)^{m+1}\cdot
f_{p},\forall~{}m,p\in\mathbb{N};$

v)

$f_{n}+f_{n+2}=l_{n+1},\forall~{}n\in\mathbb{N};$

2\. Special Fibonacci quaternions in the quaternion algebra
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$

In this section we determine the zero divisors Fibonacci quaternions,
respectively the invertible Fibonacci quaternions in the quaternion rings
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right),$
$\mathbb{H}_{\mathbb{Z}_{3}}\left(-1,-1\right)$,
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right),$
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right),$ where $p$ is a prime positive
integer, $p>5.$ The idea for to find these elements is the following: since
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right),$ (where $p$ is a prime
positive integer) is a split algebra, it is isomorphic to the matrix ring
$\textit{M}_{2}\left(\mathbb{Z}_{p}\right).$ If $\varphi$ $:$
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$$\rightarrow$$\textit{M}_{2}\left(\mathbb{Z}_{p}\right)$
is a rings isomorphism, then a Fibonacci quaternion $F_{n}$ is a zero divisor
in $\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ if and only if
$\varphi\left(F_{n}\right)$ is a zero divisor in
$\textit{M}_{2}\left(\mathbb{Z}_{p}\right)$ (see [Mi, Se; 11], Theorem 2.1).
Moreover, it is know that a matrix
$A$$\in$$\textit{M}_{2}\left(\mathbb{Z}_{p}\right)$ is a zero divisor in
$\textit{M}_{2}\left(\mathbb{Z}_{p}\right)$ if and only if it is not
invertible in $\textit{M}_{2}\left(\mathbb{Z}_{p}\right).$ These involve that,
a Fibonacci quaternion is a zero divisor in
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ if and only if it is not
invertible in $\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$ So, a Fibonacci
quaternion $F_{n}$ is a zero divisor in
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ if and only if
$n\left(F_{n}\right)=0.$

Proposition 2.1. i) Fibonacci quaternions $F_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if
$n$$\equiv$$0$ (mod $3$).
ii) In $\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ there are one Fibonacci
quaternion which is zero divisor and two Fibonacci quaternions which are
invertible.
Proof. i) According to Proposition 1.6 and Proposition 1.3 we obtain that
Fibonacci quaternions $F_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if
$f_{2n+3}$$\equiv$$0$ (mod $2$) if and only if $n$$\equiv$$0$ (mod $3$).
ii) There are $16$ elements in the quaternion algebra
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$. According to Theorem 1.2 we
know that there are $10$ zero divisors in
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right).$ We want to determine how many
of them are zero divisors Fibonacci quaternions and after this we determine
how many invertible Fibonacci quaternions are in the quaternion algebra
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right).$
Applying i) and Proposition 1.3 we obtain that there is one Fibonacci
quaternion which is zero divisor in
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right),$ namely for
$f_{n}=\overline{0},$ $f_{n+1}=\overline{1},$ $f_{n+2}=\overline{1},$
$f_{n+3}=\overline{0}$ (in $\mathbb{Z}_{2}$), so $F_{n}=i+j.$
Applying i) and Proposition 1.3, it results immediately that there are two
invertible Fibonacci quaternions in
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ : $F_{n_{1}}=\overline{1}+i+k$
and $F_{n_{2}}=\overline{1}+j+k.$
Proposition 2.2. All Fibonacci quaternions $F_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{3}}\left(-1,-1\right)$ and the number
of these elements is $8.$

Proof. Applying Proposition 1.6 we obtain that
$n\left(F_{n}\right)=\overline{0}$ in $\mathbb{Z}_{3},$
$\left(\forall\right)n$$\in$$\mathbb{N}.$ This implies that all Fibonacci
quaternions $F_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{3}}\left(-1,-1\right).$
There are $81$ elements in the quaternion algebra
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right).$ According to Theorem 1.2,
$33$ of them are zero divisors. Now, we determine how many of these $33$
elements are zero divisors Fibonacci quaternions. From the previously proved
and Proposition 1.4 we obtain the following zero divisors Fibonacci
quaternions in $\mathbb{H}_{\mathbb{Z}_{3}}\left(-1,-1\right):$
$F_{n_{1}}=i+j+\overline{2}k,$ $F_{n_{2}}=\overline{1}+i+\overline{2}j,$
$F_{n_{3}}=\overline{1}+\overline{2}i+\overline{2}k,$
$F_{n_{4}}=\overline{2}+\overline{2}j+\overline{2}k,$
$F_{n_{5}}=\overline{2}i+\overline{2}j+k,$
$F_{n_{6}}=\overline{2}+\overline{2}i+j,$ $F_{n_{7}}=\overline{2}+i+k,$
$F_{n_{8}}=\overline{1}+j+k.$

Proposition 2.3. i) Fibonacci quaternions $F_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if
$n$$\equiv$$1$ (mod $5$).
ii) In $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ there are $4$ Fibonacci
quaternions which are zero divisors and $16$ Fibonacci quaternions which are
invertible.
Proof. i) According to Proposition 1.6 and Proposition 1.5 (ii) we obtain that
Fibonacci quaternions $F_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if
$f_{2n+3}$$\equiv$$0$ ( mod $5$) if and only if $2n+3$$\equiv$$0$ (mod $5$) if
and only if $n$$\equiv$$1$ (mod $5$).
ii) There are $625$ elements in the quaternion algebra
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right).$ According to Theorem 1.2,
$145$ of them are zero divisors. Applying i) and Proposition 1.5 we obtain $4$
zero divisor Fibonacci quaternion in
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right):$
$F_{n_{1}}=\overline{1}+i+\overline{2}j+\overline{3}k,$
$F_{n_{2}}=\overline{3}+\overline{3}i+j+\overline{4}k,$
$F_{n_{3}}=\overline{4}+\overline{4}i+\overline{3}j+\overline{2}k,$
$F_{n_{4}}=\overline{2}+\overline{2}i+\overline{4}j+k.$
Applying i) and Proposition 1.5, it results immediately that there are $16$
invertible Fibonacci quaternions in
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ :
$F_{n_{1}}=i+j+\overline{2}k,$
$F_{n_{2}}=\overline{1}+\overline{2}i+\overline{3}j,$
$F_{n_{3}}=\overline{2}+\overline{3}i+\overline{3}k,$
$F_{n_{4}}=\overline{3}+\overline{3}j+\overline{3}k,$
$F_{n_{5}}=\overline{3}i+\overline{3}j+k,$
$F_{n_{6}}=\overline{3}+i+\overline{4}j,$
$F_{n_{7}}=\overline{1}+\overline{4}i+\overline{4}k,$
$F_{n_{8}}=\overline{4}+\overline{4}j+\overline{4}k,$
$F_{n_{9}}=\overline{4}i+\overline{4}j+\overline{3}k,$
$F_{n_{10}}=\overline{4}+\overline{3}i+\overline{2}j,$
$F_{n_{11}}=\overline{3}+\overline{2}i+\overline{2}k,$
$F_{n_{12}}=\overline{2}+\overline{2}j+\overline{2}k,$
$F_{n_{13}}=\overline{2}i+\overline{2}j+\overline{4}k,$
$F_{n_{14}}=\overline{2}+\overline{4}i+j,$ $F_{n_{15}}=\overline{4}+i+k,$
$F_{n_{16}}=\overline{1}+j+k.$

We asked our self what happens in the general case, when we work in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$ We obtain the
following result:
Theorem 2.1. Let $p$ be a prime positive integer, $p$$\equiv$$13,17$ (mod
$20$), $l$ be a positive integer and
$n=\frac{p\cdot\left(2l+1\right)+2l-1}{4}-1.$ Then, the Fibonacci quaternions
$F_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$

Proof. We remark that $4$$|$$\left(p\cdot\left(2l+1\right)+2l-1\right),$ so
$n$ is an integer number.
Since $p$$\equiv$$13,17$ (mod $20$), it results that the Legendre symbol
$\left(\frac{p}{5}\right)=-1$ and
$\frac{p-\left(\frac{p}{5}\right)}{2}=\frac{p+1}{2}$ is an odd integer number.
Since $n=\frac{p\cdot\left(2l+1\right)+2l-1}{4}-1,$ it results
$\frac{p+1}{2}|\left(2n+3\right).$ Applying Proposition 1.2, it results that
$f_{\frac{p+1}{2}}|f_{2n+3}.$ According to Theorem 1.1 and Proposition 1.6, we
obtain that $n\left(F_{n}\right)=\overline{0}$ in $\mathbb{Z}_{p}.$ So, if $p$
is a prime positive integer, $p$$\equiv$$13,17$ (mod $20$) and
$n=\frac{p\cdot\left(2l+1\right)+2l-1}{4}-1,$ $l$$\in$$\mathbb{N}^{*},$ then
the Fibonacci quaternions $F_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$

Now, we generalize the results from Proposition 2.1 and Proposition 2.3,
obtaining a necessary and sufficient condition for that a Fibonacci quaternion
$F_{n}$ to be a zero divisor in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$ The difference between the
expression of $n$ from Theorem 2.1 and the expression of $n$ from Theorem 2.2
is that $n$ from Theorem 2.2 depends of $z\left(p\right).$
Theorem 2.2. Let $p$ be a prime positive integer, $p>5.$ Then:
i) Fibonacci quaternions $F_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ if and only if
$n=\frac{\left(2l+1\right)\cdot z\left(p\right)-1}{2}-1,$ where
$l$$\in$$\mathbb{N}.$
ii) In the quaternion ring $\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$
there are $4$ Fibonacci quaternions which are zero divisors and
$k\left(p\right)-4$ Fibonacci quaternions which are invertible.

Proof. i) Let $F_{n}$ be a Fibonacci quaternion which is zero divisor in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right).$ Applying
Proposition 1.6, it results that $p|f_{2n+3}.$ This is equivalent with
$z\left(p\right)|\left(2n+3\right)$ (according to Remark 1.1). This is
equivalent with $z\left(p\right)$ is an odd number and $\left(\exists\right)$
$l$$\in$$\mathbb{N}$ such that $2n+3=\left(2l+1\right)\cdot z\left(p\right)$.
We obtain that $n=\frac{\left(2l+1\right)\cdot z\left(p\right)-1}{2}-1,$
$l$$\in$$\mathbb{N}.$
ii) Since $z\left(p\right)$ is an odd number, using Remark 1.3 and a reasoning
similar to the one used in Proposition 2.1 (ii) and Proposition 2.3 (ii), we
obtain that Fibonacci quaternions $F_{n}$ are zero divisors in the quaternion
ring $\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ if and only if
$f_{2n+3}$$\equiv$$0$ ( mod $p$) if and only if $2n+3$$\equiv$$0$ (mod
$z\left(p\right)$)(according to Remark 1.1) if and only if
$2n$$\equiv$$z\left(p\right)-3$ (mod $z\left(p\right)$). Since
$g.c.d.\left(2,z\left(p\right)\right)=1,$ it results that $\overline{2}$ is
invertible in $\mathbb{Z}_{z\left(p\right)},$ so, the congruence
$2n$$\equiv$$z\left(p\right)-3$ (mod $z\left(p\right)$) has unique solution
mod $z\left(p\right).$ Applying Remark 1.3, we get that in
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ there are $4$ Fibonacci
quaternions which are zero divisors and $k\left(p\right)-4$ Fibonacci
quaternions which are invertible.

3\. Generalized Fibonacci-Lucas quaternions which are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{r}}\left(-1,-1\right).$

Let $\left(f_{n}\right)_{n\geq 0}$ be the Fibonacci sequence and
$\left(l_{n}\right)_{n\geq 0}$ be the Lucas sequence.

In the paper [Fl, Sa; 15], we introduced the generalized Fibonacci-Lucas
numbers and the generalized Fibonacci-Lucas quaternions
$\left(F_{n}\right)_{n\geq 0},$ namely:
let $n,p,q$ be arbitrary integers, $n\geq 0$. The numbers
$\left(g^{p,q}_{n}\right)_{n\geq 0},$

$g^{p,q}_{n+1}=pf_{n}+ql_{n+1},\ n\geq 0$

are called the generalized Fibonacci-Lucas numbers.

Let $r$ be a prime positive integer and let
$\mathbb{H}_{\mathbb{Z}_{r}}\left(-1,-1\right)$ be the quaternion algebra over
the finite field $\mathbb{Z}_{r}$ with a basis $\left\\{1,i,j,k\right\\},$
where $i^{2}=-1,$ $j^{2}=-1,$ $k=ij=-ji.$ The element

$G^{p,q}_{n}=g^{p,q}_{n}+g^{p,q}_{n+1}\cdot i+g^{p,q}_{n+2}\cdot
j+g^{p,q}_{n+3}\cdot k$

is called the $n$-th generalized Fibonacci-Lucas quaternion (in the quaternion
algebra $\mathbb{H}_{\mathbb{Z}_{r}}\left(-1,-1\right)$).

We determine the generalized Fibonacci-Lucas quaternions which are zero
divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{r}}\left(-1,-1\right),$ when $r=2$ or $r=3$ or $r=5$
or $r=p$ or $r=q.$
First, we compute the norm for the $n$-th generalized Fibonacci-Lucas
quaternion in $\mathbb{H}_{\mathbb{Z}_{r}}\left(-1,-1\right).$

Proposition 3.1. Let $n,p,q$ be arbitrary integers, $n\geq 0$. Let
$G^{p,q}_{n}$ be the $n$-th generalized Fibonacci-Lucas quaternion. Let $r$ be
a prime positive integer. Then, the norm of $G^{p,q}_{n}$ in the quaternion
algebra $\mathbb{H}_{\mathbb{Z}_{r}}\left(-1,-1\right),$ is given by

$\textbf{n}\left(G^{p,q}_{n}\right)=\overline{3p^{2}f_{2n+1}+15q^{2}f_{2n+3}+6pq\left(f_{2n+1}+f_{2n+3}\right)}\
(in\ \mathbb{Z}_{r}).$

Proof.

$n\left(G^{p,q}_{n}\right)=G^{p,q}_{n}\overline{G^{p,q}_{n}}=\left(g^{p,q}_{n}\right)^{2}+\left(g^{p,q}_{n+1}\right)^{2}+\left(g^{p,q}_{n+2}\right)^{2}+\left(g^{p,q}_{n+3}\right)^{2}=$
$=\left(pf_{n-1}+ql_{n}\right)^{2}+\left(pf_{n}+ql_{n+1}\right)^{2}+\left(pf_{n+1}+ql_{n+2}\right)^{2}+\left(pf_{n+2}+ql_{n+3}\right)^{2}=$
$=p^{2}\left(f^{2}_{n-1}+f^{2}_{n}+f^{2}_{n+1}+f^{2}_{n+2}\right)+q^{2}\left(l^{2}_{n}+l^{2}_{n+1}+l^{2}_{n+2}+l^{2}_{n+3}\right)+$
$+2pq\left(f_{n-1}l_{n}+f_{n}l_{n+1}+f_{n+1}l_{n+2}+f_{n+2}l_{n+3}\right).$

Applying Proposition 1.7 (i, iii, iv), we have:

$n\left(G^{p,q}_{n}\right)=p^{2}\left(f_{2n-1}+f_{2n+3}\right)+q^{2}\left(5f_{2n+1}+5f_{2n+5}\right)+$
$+2pq\left[f_{2n-1}+\left(-1\right)^{n}\cdot
f_{1}+f_{2n+1}+\left(-1\right)^{n+1}\cdot
f_{1}+f_{2n+3}+\left(-1\right)^{n+2}\cdot
f_{1}+f_{2n+5}+\left(-1\right)^{n+3}\cdot f_{1}\right].$

Using Proposition 1.7 (ii), we obtain:

$\textbf{n}\left(G^{p,q}_{n}\right)=\overline{3p^{2}f_{2n+1}+15q^{2}f_{2n+3}+6pq\left(f_{2n+1}+f_{2n+3}\right)}\
(in\ \mathbb{Z}_{r}).$

Proposition 3.2. Let $p,q$ be two integers and let $n$ be an arbitrary
positive integer. Then, the following statements are true:
i) if $p$$\equiv$$0$ (mod $2$) and $q$$\equiv$$1$ (mod $2$), then the
generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if
$n$$\equiv$$0$ (mod $3$);
ii) if $p$$\equiv$$1$ (mod $2$) and $q$$\equiv$$0$ (mod $2$) then the
generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if
$n$$\equiv$$1$ (mod $3$);
iii) if $p;q$$\equiv$$0$ (mod $2$) then all generalized Fibonacci-Lucas
quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right);$
iv) if $p;q$$\equiv$$1$ (mod $2$), then the generalized Fibonacci-Lucas
quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if $n$$\equiv$$2$
(mod $3$);

Proof. i) The generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero
divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if
$n\left(G^{p,q}_{n}\right)=\overline{0}$ (in $\mathbb{Z}_{2}$ ). Applying
Proposition 3.1 and the facts that $r=2$ and $p$$\equiv$$0$ (mod $2$) and
$q$$\equiv$$1$ (mod $2$), it results that
$n\left(G^{p,q}_{n}\right)=\overline{f_{2n+3}}$ in $\mathbb{Z}_{2}.$ Applying
Proposition 1.3, $\overline{f_{2n+3}}=\overline{0}$ in $\mathbb{Z}_{2}$ if and
only if $n$$\equiv$$0$ (mod $3$).
ii) If $p$$\equiv$$1$ (mod $2$) and $q$$\equiv$$0$ (mod $2$), similarly as in
i) we get that the generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are
zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if
$\overline{f_{2n+1}}=\overline{0}$ in $\mathbb{Z}_{2}$ if and only if
$n$$\equiv$$1$ (mod $3$).
iii) If $p;q$$\equiv$$0$ (mod $2$), it results that
$n\left(G^{p,q}_{n}\right)=\overline{0}$ (in $\mathbb{Z}_{2}$) for
$\left(\forall\right)$ $G^{p,q}_{n},$ so, we obtain immediate that all
generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right).$
iv) If $p;q$$\equiv$$1$ (mod $2$), applying Proposition 1.7 (v) it results
that
$n\left(G^{p,q}_{n}\right)=\overline{f_{2n+1}}+\overline{f_{2n+3}}=\overline{l_{2n+2}}$
(in $\mathbb{Z}_{2}$ ). Using a reasoning as in i) and Proposition 1.3 (iii),
we obtain that the generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are
zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{2}}\left(-1,-1\right)$ if and only if
$\overline{l_{2n+2}}=\overline{0}$ (in $\mathbb{Z}_{2}$) if and only if
$n$$\equiv$$2$ (mod $3$).

Proposition 3.3. Let $p,q$ be two integers and let $n$ be an arbitrary
positive integer. Then, all generalized Fibonacci-Lucas quaternions
$G^{p,q}_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{3}}\left(-1,-1\right).$

Proof. Applying Proposition 3.1 we obtain that
$n\left(G^{p,q}_{n}\right)=\overline{0}$ in $\mathbb{Z}_{3},$
$\left(\forall\right)n$$\in$$\mathbb{N}.$ This implies that all generalized
Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion
ring $\mathbb{H}_{\mathbb{Z}_{3}}\left(-1,-1\right).$

Proposition 3.4. Let $p,q$ be two integers and let $n$ be an arbitrary
positive integer. Then, the following statements are true:
i) if $p$$\equiv$$0$ (mod $5$), then all generalized Fibonacci-Lucas
quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$;
ii) if $5$ does not divide $p$ and $p$$\equiv$$q$ (mod $5$), then the
generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if
$n$$\equiv$$4$ (mod $5$);
iii) if $5$ does not divide $p$ and $p+q$$\equiv$$0$ (mod $5$), then the
generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if
$n$$\equiv$$0$ (mod $5$);
iv) if $p$$\equiv$$1$ (mod $5$), $q$$\equiv$$2$ (mod $5$) or if $p$$\equiv$$3$
(mod $5$), $q$$\equiv$$1$ (mod $5$) or if $p$$\equiv$$2$ (mod $5$),
$q$$\equiv$$4$ (mod $5$) or if $p$$\equiv$$4$ (mod $5$), $q$$\equiv$$3$ (mod
$5$),then the generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero
divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if $n$$\equiv$$1$
(mod $5$);
v) if $p$$\equiv$$2$ (mod $5$), $q$$\equiv$$1$ (mod $5$) or if $p$$\equiv$$3$
(mod $5$), $q$$\equiv$$4$ (mod $5$) or if $p$$\equiv$$4$ (mod $5$),
$q$$\equiv$$2$ (mod $5$) or if $p$$\equiv$$1$ (mod $5$), $q$$\equiv$$3$ (mod
$5$), then the generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero
divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if $n$$\equiv$$3$
(mod $5$).
vi) if $5$ does not divide $p$ and $q$$\equiv$$0$ (mod $5$), then the
generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the
quaternion ring $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if
$n$$\equiv$$2$ (mod $5$).

Proof. i) If $p$$\equiv$$0$ (mod $5$) according to Proposition 3.1 we obtain
that $n\left(G^{p,q}_{n}\right)=\overline{0}$ in $\mathbb{Z}_{5},$
$\left(\forall\right)n$$\in$$\mathbb{N}.$ This implies that all generalized
Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion
ring $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right).$
ii) Applying Proposition 3.1, we have:
$n\left(G^{p,q}_{n}\right)=\overline{3p^{2}f_{2n+1}+6pq\left(f_{2n+1}+f_{2n+3}\right)}$
in $\mathbb{Z}_{5}.$ Using the recurrence of Fibonacci sequence, it results
that:

$n\left(G^{p,q}_{n}\right)=\overline{p\cdot\left[\left(3p+2q\right)f_{2n+1}+qf_{2n+2}\right]}\
in\;\mathbb{Z}_{5}.$

The generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in
the quaternion ring $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and
only if $n\left(G^{p,q}_{n}\right)=\overline{0}$ (in $\mathbb{Z}_{5}$). Since
$5$ does not divide $p,$ the $n\left(G^{p,q}_{n}\right)=\overline{0}$ (in
$\mathbb{Z}_{5}$) if and only if
$\left(3p+2q\right)f_{2n+1}+qf_{2n+2}$$\equiv$$0$ (mod $5$). Since
$p$$\equiv$$q$ (mod $5$), we obtain that the congruence
$\left(3p+2q\right)f_{2n+1}+qf_{2n+2}$$\equiv 0$ (mod $5$) is equivalent with
$qf_{2n+2}$$\equiv 0$ (mod $5$) if and only in $f_{2n+2}$$\equiv 0$ (mod $5$).
According to Proposition 1.5, the last congruence is equivalent with
$2n+2$$\equiv 0$ (mod $5$) if and only if $n$$\equiv 4$ (mod $5$).
iii) If $5$ does not divide $p$, similarly as in ii) the generalized
Fibonacci-Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion
ring $\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if
$\left(3p+2q\right)f_{2n+1}+qf_{2n+2}$$\equiv 0$ (mod $5$). Using the fact
that $p+q$$\equiv$$0$ (mod $5$) and the recurrence of Fibonacci sequence, the
last congruence is equivalent with $qf_{2n}$$\equiv 0$ (mod $5$) if and only
if $f_{2n}$$\equiv 0$ (mod $5$) if and only if $n$$\equiv 0$ (mod $5$)
(according to Proposition 1.5).
iv) Similarly as in ii), the generalized Fibonacci-Lucas quaternions
$G^{p,q}_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if
$\left(3p+2q\right)f_{2n+1}+qf_{2n+2}$$\equiv 0$ (mod $5$). We denote
$E=\left(3p+2q\right)f_{2n+1}+qf_{2n+2}.$ We have:

$E\equiv\left\\{\begin{array}[]{c}2f_{2n+1}+2f_{2n+2}\;(mod\;5),\ if\ p\equiv
1\;(mod\;5),\ q\equiv 2\;(mod\;5)\\\ f_{2n+1}+f_{2n+2}\;(mod\;5),\ if\ p\equiv
3\;(mod\;5),\ q\equiv 1\;(mod\;5)\\\ 4f_{2n+1}+4f_{2n+2}\;(mod\;5),\ if\
p\equiv 2\;(mod\;5),\ q\equiv 4\;(mod\;5)\\\ 3f_{2n+1}+3f_{2n+2}\;(mod\;5),\
if\ p\equiv 4\;(mod\;5),\ q\equiv 3\;(mod\;5).\\\ \end{array}\right.$

This is equivalent with:

$E\equiv\left\\{\begin{array}[]{c}2f_{2n+3}\;(mod\;5),\ if\ p\equiv
1\;(mod\;5),\ q\equiv 2\;(mod\;5)\\\ f_{2n+3}\;(mod\;5),\ if\ p\equiv
3\;(mod\;5),\ q\equiv 1\;(mod\;5)\\\ 4f_{2n+3}\;(mod\;5),\ if\ p\equiv
2\;(mod\;5),\ q\equiv 4\;(mod\;5)\\\ 3f_{2n+3}\;(mod\;5),\ if\ p\equiv
4\;(mod\;5),\ q\equiv 3\;(mod\;5).\\\ \end{array}\right.$

For to have $E\equiv$ $0$ (mod $5$) it is necessary and sufficient to have
$f_{2n+3}\equiv$$0$ (mod $5$). According to Proposition 1.5, the last
congruence is equivalent with $2n+3$$\equiv 0$ (mod $5$) if and only if
$n$$\equiv 1$ (mod $5$).
v) Similarly as in iv), we have:

$E\equiv\left\\{\begin{array}[]{c}3f_{2n+1}+f_{2n+2}\;(mod\;5),\ if\ p\equiv
2\;(mod\;5),\ q\equiv 1\;(mod\;5)\\\ 2f_{2n+1}+4f_{2n+2}\;(mod\;5),\ if\
p\equiv 3\;(mod\;5),\ q\equiv 4\;(mod\;5)\\\ f_{2n+1}+2f_{2n+2}\;(mod\;5),\
if\ p\equiv 4\;(mod\;5),\ q\equiv 2\;(mod\;5)\\\
4f_{2n+1}+3f_{2n+2}\;(mod\;5),\ if\ p\equiv 1\;(mod\;5),\ q\equiv
3\;(mod\;5).\\\ \end{array}\right.$

Using Fibonacci recurrence, we obtain:

$E\equiv\left\\{\begin{array}[]{c}f_{2n}+4f_{2n+1}\;(mod\;5),\ if\ p\equiv
2\;(mod\;5),\ q\equiv 1\;(mod\;5)\\\ 2f_{2n+3}+2f_{2n+2}\;(mod\;5),\ if\
p\equiv 3\;(mod\;5),\ q\equiv 4\;(mod\;5)\\\ f_{2n+3}+f_{2n+2}\;(mod\;5),\ if\
p\equiv 4\;(mod\;5),\ q\equiv 2\;(mod\;5)\\\ -f_{2n+1}-2f_{2n+2}\;(mod\;5),\
if\ p\equiv 1\;(mod\;5),\ q\equiv 3\;(mod\;5).\\\ \end{array}\right.$

This is equivalent with:

$E\equiv\left\\{\begin{array}[]{c}4f_{2n-1}\;(mod\;5),\ if\ p\equiv
2\;(mod\;5),\ q\equiv 1\;(mod\;5)\\\ 2f_{2n+4}\;(mod\;5),\ if\ p\equiv
3\;(mod\;5),\ q\equiv 4\;(mod\;5)\\\ f_{2n+4}\;(mod\;5),\ if\ p\equiv
4\;(mod\;5),\ q\equiv 2\;(mod\;5)\\\ -f_{2n+4}\;(mod\;5),\ if\ p\equiv
1\;(mod\;5),\ q\equiv 3\;(mod\;5).\\\ \end{array}\right.$

Similarly as in ii), the generalized Fibonacci-Lucas quaternions $G^{p,q}_{n}$
are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if $E$$\equiv 0$
(mod $5$). This is equivalent with $f_{2n-1}\equiv$$0$ (mod $5$), respectively
$f_{2n+4}\equiv$$0$ (mod $5$). According to Proposition 1.5, each of these two
congruences is equivalent to $n$$\equiv$$3$ (mod $5$).
vi) We get $E$$\equiv$$3pf_{2n+1}$ (mod $5$). For to have $E\equiv$$0$ (mod
$5$) it is necessary and sufficient to have $f_{2n+1}\equiv$$0$ (mod $5$).
According to Proposition 1.5, the last congruence is equivalent with
$2n+1$$\equiv 0$ (mod $5$) if and only if $n$$\equiv 2$ (mod $5$).

Now, we consider $r=p\neq q,$ $p>5$ and we obtain the following result:

Theorem 3.1. Let $p,q$ be two prime positive integers, $p\neq q,$ $p>5$ and
let $n$ be an arbitrary positive integer. Then, the generalized Fibonacci-
Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{p}}\left(-1,-1\right)$ if and only if
$n=\frac{\left(2l+1\right)\cdot z\left(p\right)-1}{2}-1,$ where
$l$$\in$$\mathbb{N}.$

Proof. Applying Proposition 3.1 and the fact that $r=p,$ it results that
$\textbf{n}\left(G^{p,q}_{n}\right)=\overline{15q^{2}f_{2n+3}}$ in
$\mathbb{Z}_{p}.$ Since $p,q$ are primes, $p\neq q,$ $p>5,$ we obtain that
$\textbf{n}\left(G^{p,q}_{n}\right)=\overline{0}$ (in $\mathbb{Z}_{p}$) if and
only if $p|f_{2n+3}.$ Similarly with the proof of Theorem 2.2 we obtain that
$p|f_{2n+3}$ is equivalent with $n=\frac{\left(2l+1\right)\cdot
z\left(p\right)-1}{2}-1,$ where $l$$\in$$\mathbb{N}.$

Now we consider $r=q\neq p,$ $q>5$ and we obtain the following result:

Theorem 3.2. Let $p,q$ be two prime positive integers, $p\neq q,$ $q>5$ and
let $n$ be an arbitrary positive integer. Then, the generalized Fibonacci-
Lucas quaternions $G^{p,q}_{n}$ are zero divisors in the quaternion ring
$\mathbb{H}_{\mathbb{Z}_{q}}\left(-1,-1\right)$ if and only if
$n=\frac{\left(2l+1\right)\cdot z\left(q\right)-1}{2},$ where
$l$$\in$$\mathbb{N}.$

Proof. Applying Proposition 3.1 and the fact that $r=q,$ it results that
$\textbf{n}\left(G^{p,q}_{n}\right)=\overline{3p^{2}f_{2n+1}}$ in
$\mathbb{Z}_{q}.$ Since $p,q$ are primes, $p\neq q,$ $q>5,$ we obtain that
$\textbf{n}\left(G^{p,q}_{n}\right)=\overline{0}$ (in $\mathbb{Z}_{q}$) if and
only if $q|f_{2n+1}.$ Similarly with the proof of Theorem 2.2 we obtain that
$q|f_{2n+1}$ is equivalent with $n=\frac{\left(2l+1\right)\cdot
z\left(q\right)-1}{2},$ where $l$$\in$$\mathbb{N}.$

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Diana SAVIN

Faculty of Mathematics and Computer Science,

Ovidius University,

Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA

http://www.univ-ovidius.ro/math/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>