# Rule-Based, Neural and LLM Back-Translation:
Comparative Insights from a Variant of Ladin

Samuel Frontull  Georg Moser
Department of Computer Science
University of Innsbruck, Innsbruck, Austria
{samuel.frontull<EMAIL_ADDRESS>
###### Abstract

This paper explores the impact of different back-translation approaches on
machine translation for Ladin, specifically the Val Badia variant. Given the
limited amount of parallel data available for this language (only 18k
Ladin–Italian sentence pairs), we investigate the performance of a
multilingual neural machine translation model fine-tuned for Ladin–Italian. In
addition to the available authentic data, we synthesise further translations
by using three different models: a fine-tuned neural model, a rule-based
system developed specifically for this language pair, and a large language
model. Our experiments show that all approaches achieve comparable translation
quality in this low-resource scenario, yet round-trip translations highlight
differences in model performance.

Rule-Based, Neural and LLM Back-Translation:
Comparative Insights from a Variant of Ladin

Samuel Frontull and Georg Moser Department of Computer Science University of
Innsbruck, Innsbruck, Austria {samuel.frontull<EMAIL_ADDRESS>

##  1 Introduction

In recent years, a variety of methods have been developed to apply neural
machine translation (NMT) also in low-resource scenarios Shi et al. (2022);
Haddow et al. (2022); Ranathunga et al. (2023). The back-translation technique
has shown to be particularly effective in such settings Sennrich et al.
(2016); Edunov et al. (2018), offering the potential for substantial
improvements in translation quality.

This work investigates the influence of the back-translation model selection
for a low-resource language. We do this, by comparing the results obtained by
fine-tuning a pre-trained multilingual NMT model using synthesised
translations generated by  (i) a NMT system fine-tuned on the available
parallel data,  (ii) a rule-based machine translation (RBMT) system developed
for the specific language pair, and  (iii) a large language model (LLM)
prompted to translate given texts, accompanied by 8 exemplary samples.

The quality of the synthesised data, which in turn is determined by the
underlying models used to generate it, matters Burlot and Yvon (2018). In our
case, the synthesised translations originate from three models based on a
different paradigm. Thus, the synthesised data is characterised by the
specific strengths and weaknesses of the respective paradigms.

Rule-based systems are robust and computationally lightweight, but may face
challenges in dealing with ambiguity. Moreover, they lag behind at the
grammatical level. Neural models show a high ability to adapt to provided
texts, but perform less well when confronted with out-of-domain data Shen et
al. (2021). In contrast, language model-based approaches (LLMs) are praised
for their ability to produce fluent, coherent texts, but they are prone to
hallucinations Rawte et al. (2023). It is therefore an interesting question to
investigate how this affects the quality of the NMT models trained on this
data. This comparative analysis sheds light on the nuanced contrasts inherent
in the different MT methods.

Our results show that in this low-resource scenario the back-translation model
does not have a significant impact, and the performance of the models
converges to similar results in terms of BLEU/chrF++ points. This assertion is
supported by an empirical analysis carried out on the Val Badia variant of
Ladin. Our main contributions are:

  * •

we are the first to explore MT for Ladin in general, with a specific focus on
the Val Badia variant,

  * •

we compare an RBMT-, an NMT- and an LLM-based back-translation, providing
insights into the efficacy of the methods for Ladin,

  * •

we establish baseline results and make the test data, the RBMT system, as well
as the best-performing models publicly available

In Section 2 we describe our data collection and corpus creation process. In
Section 3 we present the three different methods used for the back-translation
of monolingual Ladin data into Italian. Section 4 gives an overview of the
conducted experiments and Section 5 presents the obtained results. Section 6
discusses related work and similar approaches. In Section 7, we summarize and
discuss future work.

#### The Ladin Language

Ladin is an officially recognised minority language, and thus taught in
schools, used in the media, and employed in public administration. For this
reason, an effective machine translation system could make a significant
contribution to facilitating and supporting communication in this language.
However, Ladin is still an unexplored language in the field of machine
translation. Indeed, nearly no parallel data111In a machine readable format.
is publicly available for this language, except for a few hundred samples on
OPUS Tiedemann (2012). This language, spoken by around 30,000 people in the
northern Italian Dolomite regions, exhibits significant diversity across its
five main variants (_Val Badia_ , _Gherdëina_ , _Fascia_ , _Fodom_ ,
_Anpezo_), each shaped uniquely by its development in different valleys. This
diversity is not only evident in the spoken language but has also resulted in
distinct standards for written communication. The first author of the paper
originates from the _Val Badia_ and is a native speaker of Ladin. Therefore,
in this work we concentrate on the standard written language of this valley.
In the rest of the paper, we will use _lvb_ as language code to refer to this
variant of Ladin, and _ita_ for Italian.

##  2 Data

This section gives an overview of the linguistic resources available for the
Ladin language and describes the method employed to collect data for the
specific Val Badia variant of Ladin.

###  2.1 Available Resources

Publicly accessible parallel data for Ladin is scarce. The Open Parallel
Corpus Tiedemann (2012) e.g. lists 1543 Ladin–German, 220 Ladin–Italian, and
81 Ladin–English sentences. However, these texts are mainly specific to the
variants of Gherdëina and Fassa and were not disseminated by public
institutions. For our experiments, we were provided with the archive of the
weekly newspaper _La Usc di Ladins_ 222https://www.lausc.it/ and a digitised
version of the dictionary Ladin Val Badia – Italian Moling et al. (2016). From
these data sources we extracted monolingual texts as well as a small dataset
of parallel sentences. We furthermore used the dictionary as the basis for
implementing a RBMT system. The collection of other parallel texts is time-
consuming and has therefore been left for future work.

###  2.2 Parallel Data

The Ladin (Val Badia) – Italian dictionary Moling et al. (2016) contains,
alongside the word entries, also sentences that illustrate their usage. For
these sentences the corresponding Italian translation is also given. We have
collected this data to create our training dataset, which contains a total of
$18,139$ sentences. These sentences are basic and short because they were
created specifically to illustrate the use of words and phrases. The average
length is $23.43$ and $25.69$ characters for Ladin and Italian respectively.
This dataset has been publicly
released.333https://www.doi.org/10.57967/hf/1878

###  2.3 Ladin Monolingual Data

The Ladin newspaper _La Usc di Ladins_ , digitally archived since 2012,
provides an extensive dataset of monolingual texts. These texts are published
in five different variants, each corresponding to one of the five Ladin
valleys. We extracted these texts from the PDF documents and segmented them
into individual sentences using the NLTK library Bird et al. (2009),
specifically setting Italian as the language to accommodate Ladin. In total,
we accumulated $1,937,608$ sentences. These sentences had to be categorised by
variant, as described below.

#### Variant Classification

In order to train a variant classifier, labeled training data is essential.
However, the monolingual data from the newspaper PDFs lacked these labels.
Therefore, we collected the texts from the newspaper’s
website.444https://www.lausc.it Here, the article excerpts are categorized
according to their origin valleys and the corresponding language variants,
allowing us to create a labeled dataset.

We gathered a corpus of $7,766$ article excerpts with a total of $42,745$
individual sentences for training. These sentences were then split into
training (comprising $75\%$ of the sentences) and test data (the remaining
$25\%$). Using the $2,500$ most frequent $3$-gram characters as features, we
trained an XGBoost variant-classifier Chen and Guestrin (2016). On the test
data, our classifier achieved $94.48\%$ accuracy in classifying these 5
labels.

The resulting model was used to predict the variant of each of the $1,937,608$
sentences in the monolingual dataset. Table 1 reports the respective number of
classifications (and the total number of characters) for each variant. 746,704
sentences were classified as val-badia and were considered for further
processing.

variant | # sentences | # characters
---|---|---
val-badia | $746.704$ | $71.619.515$
gherdeina | $491.575$ | $57.704.414$
fascia | $407.605$ | $52.504.357$
fodom | $146.049$ | $16.615.059$
anpezo | $145.674$ | $16.425.301$
Table 1:  Variant classification of monolingual data.

#### Data Preparation

Because of the spelling reform in 2015, we further processed the sentences
classified as val-badia to exclude any with words that are no longer valid. To
do this, we used the implementation of our RBMT system which is explained in
more detail in Section 3.2. We used the system to identify unknown words and
tried to adapt them to the new spelling according to certain rules. Sentences
where this was not possible were left out. This process ensured that the
filtered sentences fully adhered to the new spelling, which also facilitated
the rule-based translations. We collected a total of 274,665 sentences
($\approx 31\%$ of the extracted sentences) which constitute the monolingual
Ladin data we used in our experiments. Among the unused sentences, $\approx
100k$ contain only one unknown word/typo so there would be still potential to
acquire additional data if additional time were spent analysing and preparing
these texts.

###  2.4 Italian Monolingual Data

As monolingual data for Italian, we used the ELRC-CORDIS_News
dataset555https://elrc-share.eu/ from OPUS Tiedemann (2012), which contains
123,691 Italian sentences.

###  2.5 Test Data

This section introduces the three test sets on which the models were
evaluated. This test data differs considerably from the training data, so that
it can be considered out-of-domain data.

#### Testset 1

This dataset includes the statute of the _Stiftung Südtiroler Sparkasse_ , a
nonprofit foundation dedicated to supporting and promoting various initiatives
and projects, primarily within the province of Bolzano. The document is rich
in formal and legal terminology. It contains 424
sentences666https://huggingface.co/datasets/sfrontull/stiftungsparkasse-
lld_valbadia-ita.

#### Testset 2

This dataset is a festive compendium of the history of the region associated
with this language Kager (2022). It combines historical narratives with legal
and administrative statements. The result is a mixture of stylistic elements
and lexical domains. It contains 833
sentences777https://huggingface.co/datasets/sfrontull/autonomia-lld_valbadia-
ita.

#### Testset 3

This dataset delves into the literary realm with the classic story of
Pinocchio Collodi (2017), a text rich in narrative prose, dialogue and
idiomatic expressions, challenging the models with its creative and figurative
language. It contains 1563
sentences888https://huggingface.co/datasets/sfrontull/pinocchio-lld_valbadia-
ita.

##  3 Back-translation Strategies

The so-called _back-translation_ , first introduced in Sennrich et al. (2016),
refers to the process of automatic translation of monolingual texts in the
target language to the source language. This method of enriching additional
training data in the source-to-target translation direction (where the target
side remains authentic) has proven to be particularly effective and is
particularly valuable in low-resource scenarios. In this section we present
the three different back-translation strategies used in our research to
translate monolingual Ladin texts into Italian.

###  3.1 Neural MT

There is evidence that low-resource languages benefit from multilingual models
Aharoni et al. (2019). For this reason, we opted to utilise a pre-trained,
multilingual model, specifically the Helsinki-NLP/opus-mt-ine-
ine999https://huggingface.co/Helsinki-NLP/opus-mt-ine-ine model available from
the Hugging Face Model Hub, as our base model. This model, which is part of
OPUS-MT Tiedemann and Thottingal (2020), was trained to translate between 135
Indo-European languages, to which Ladin and Italian also belong.

The Marian MT model, configured for Helsinki-NLP/opus-mt-ine-ine, features 6
encoder and 6 decoder layers, each with 8 attention heads and a feed-forward
dimension of 2048. The model employs a beam search size of 6, a dropout rate
of 0.1, and an embedding size of 512. It shares embeddings between the encoder
and decoder.

We fine-tuned this model for the two translation directions _lvb_
$\rightarrow$ _ita_ and _ita_ $\rightarrow$ _lvb_ on the available authentic
training data. We trained a single model for both directions by using the tags
>>ita<< for _lvb_ $\rightarrow$ _ita_ and >>lld_Latn<< 101010We (re)used the
tag >>lld_Latn<< because it is listed as a valid target language ID, as few
Ladin texts were already included in the training of this model. for the
opposite direction as prefixes of the source text. In the rest of the paper,
we refer to this fine-tuned model as N1. For fine-tuning, we utilized the
AdamW
optimizer111111https://huggingface.co/docs/transformers/v4.41.0/en/main_classes/optimizer_schedules#transformers.AdamW
with the defaults settings.

The fine-tuning greatly improves the model in both translation directions, as
the scores reported in Table 4 and 3 show. This demonstrates that the data is
reliable and that the model adapts well.

###  3.2 Rule-based MT

For low-resource languages, RBMT frameworks offer a crucial advantage:
leveraging linguistic expertise to overcome the limitations of data-driven
methods Khanna et al. (2021). Considering the similar sentence structure and
composition of Ladin and Italian (they are both Romance languages), it can be
assumed that a rule-based MT system can also perform well without excessive
structural transfer work. The available Ladin Val Badia-Italian dictionary
served as the foundation for the rule-based MT system we developed in Apertium
Forcada and Tyers (2016) for this language pair.

This dictionary provides, in addition to the individual words and word
translations, also a list of all inflected forms for each lemma. To
effectively utilise this dictionary within our translation system, we mapped
the lexicographical data to paradigms within the framework of Apertium
(monodix format). Specifically, we created 742 paradigms for a total of 19,034
lemmas. This extensive set includes multiple lexical categories: 597 adverbs,
3,366 adjectives, 11,496 nouns, 162 pronouns, and 2,439 verbs. Additionally,
we incorporated proper nouns, short phrases, and wordgrams that were
identified during the monolingual text extraction process. The resulting
bilingual dictionary contains a total of 30,468 entries. The integration with
Apertium was facilitated by connecting to and reusing the pre-existing module
for Italian121212https://github.com/apertium/apertium-ita. The Ladin
module131313https://github.com/schtailmuel/apertium-lld-ita and the
Ladin–Italian141414https://github.com/schtailmuel/apertium-lld module can be
found on GitHub. In the rest of the paper, we refer to this RBMT system as R1.

According to aq-covtest151515https://wikis.swarthmore.edu/ling073/Apertium-
quality R1 has a coverage of $96.66\%$ on Testset 1, $95.81\%$ on Testset 2
and $95.90\%$ on Testset 3. However, since we did not develop disambiguation
modules, we designed the system to select the first suggestion in cases of
morphological and lexical ambiguity, which can sometimes result in incorrect
choices that may distort the meaning of the texts. To counteract this, and to
further enhance the rule-based translation system, we extracted the $900$ most
common word $n$-grams from the texts and added their corresponding
translations as entries to the bilingual dictionary.

In addition to the data, we have also included 13 1-level structural transfer
rules to avoid common errors. For example, in Ladin, the word _pa_ is used to
emphasize a question. In Italian, however, there is no corresponding word for
this purpose. We have therefore developed a rule to exclude this word from the
translation. The other rules include gender correction, dealing with reflexive
verbs and prepositions.

###  3.3 MT with a Large Language Model

LLMs have shown remarkable capability in understanding and generating human-
like text across various languages and domains Brown et al. (2020). However,
their performance in MT tasks exhibits significant variability across
languages, especially when comparing high-resource languages to low-resource
languages Robinson et al. (2023). We explore the utilisation of a LLM,
specifically _GPT-3.5 Turbo_ OpenAI (2024), to generate translations from
Ladin to Italian. The process involved leveraging the advanced capabilities of
the LLM, accessed through the gpt-3.5-turbo-0125 API endpoint. In the rest of
the paper, we refer to this LLM as L1.

To enhance throughput and reduce the number of API requests, we generated the
translation of 16 Ladin texts in a single request. We provided a set of 8
example translations in JSON format, randomly selected from the available
authentic training data and instructed the LLM to generate translations for 16
Ladin texts, which were also provided as a JSON dictionary, with empty Italian
translations. Listing 1 (Appendix A) showcases an exemplary prompt.

With this prompting approach, we translated the entire monolingual Ladin
corpus into Italian. By providing the exemplary translations as JSON, we were
able to reduce the failure rate (invalid/incomplete answers). The extent to
which these examples also helped with the translation itself remains open. The
entire process spanned approximately 100 hours, with an average processing
time of around 22 seconds per request.

##  4 Experiments

We used the opus-mt-ine-ine161616https://huggingface.co/Helsinki-NLP/opus-mt-
ine-ine model as base model for the experiments. In the rest of the paper, we
use BM to refer to this model. We fine-tuned BM with the various data sets
using the Transformers library Wolf et al. (2020), specifically leveraging the
Seq2SeqTrainer module. We always trained a single model for both directions
using the corresponding prefixes.

We configured the training to process batches of 16 samples, and restricted
the input and output sequences to a maximum of 128 tokens to ensure manageable
computation loads. The models were evaluated each 16,000 steps. As a stopping
criterion, we used three consecutive evaluations resulting in an improvement
of less than $0.2$ chrF points on the validation set. For training, we
utilised an NVIDIA TITAN RTX graphics card with 24 GB. In total, we have
trained 15 models:

  * •

Model N1: BM fine-tuned with the available parallel data consisting of 18,139
sentences.

  * •

Models N2/R2/L2: BM fine-tuned with authentic data and Ladin monolingual data
backtranslated (BT) to Italian using N1/R1/L1 respectively.

  * •

Models N3/R3/L3: This iteration extends the training base of N2/R2/L2 by
integrating Italian monolingual data that has been translated into Ladin
utilising N2/R2/L2 respectively.

  * •

Models N4/R4/L4: BM fine-tuned with same training data as N3/R3/L3 models, but
with Ladin and Italian monolingual data backtranslated with N3/R3/L3 model.

  * •

Models N5/R5/L5: This iteration extends the training base of N4/R4/L4 by
adding also the forward-translations (FT) as training data.

  * •

Models A1/A2: A1 was trained on the combined training data used to trained N4,
R4 and L4. In A2 we additionally included the forward-translations into the
training data.

| Testset 1 | Testset 2 | Testset 3
---|---|---|---
ref | $425.7$ | $306.3$ | $697.4$
BM | $545.8$ | $325.8$ | $595.7$
N1 | $1237.6$ | $437.8$ | $805.3$
N2 | $633.3$ | $414.0$ | $695.1$
N3 | ${484.5}$ | $331.8$ | $606.8$
N4 | $367.5$ | $323.4$ | $605.4$
N5 | $476.2$ | ${320.9}$ | ${593.4}$
R1 | $559.5$ | $421.5$ | $727.8$
R2 | $593.8$ | $405.7$ | $722.2$
R3 | $434.8$ | $309.5$ | $601.0$
R4 | $402.4$ | ${305.3}$ | ${594.3}$
R5 | ${387.8}$ | $306.1$ | $608.1$
L1 | ${{380.3}}$ | $294.3$ | ${{517.8}}$
L2 | $695.6$ | ${{406.1}}$ | $675.3$
L3 | $396.0$ | $345.4$ | $634.4$
L4 | $377.0$ | $318.0$ | $563.7$
L5 | $393.3$ | $316.1$ | $569.6$
Table 2:  Mean perplexity (_ita_) of selected models.

We refer to the models N1, …, N5 that were trained with NMT backtranslated
data as N-models. Analogously, we use the term R-models and L-models to refer
to RBMT and LLM models, respectively.

Models N4/R4/L4 illustrate the gains achieved through iterative back-
translation Hoang et al. (2018). Additionally, models N5/R5/L5 demonstrate
potential improvements achievable with synthetically generated forward-
translation data.

We evaluated these models on the 3 test sets presented in Section 2.5. The
results are presented and analysed in the following section.

|  | Testset 1 | Testset 2 | Testset 3
---|---|---|---|---
Ladin (Val Badia) $\rightarrow$ Italian |  | BLEU / chrF++ | BLEU / chrF++ | BLEU / chrF++
NMT opus-mt-ine-ine | BM | $8.17/34.81$ | $8.07/34.27$ | $2.29/21.12$
BM fine-tuned with authentic data | N1 | $12.65/41.55$ | $11.49/39.90$ | $11.83/36.40$
$+$ _lvb_ monolingual BT with N1 | N2 | $13.01/42.98$ | $12.40/41.26$ | $13.23/36.84$
$+$ _ita_ monolingual BT with N2 | N3 | $21.98/50.32$ | $19.37/47.35$ | $15.01/39.15$
$+$ _lvb_ and _ita_ monolingual BT with N3 | N4 | $\underline{22.90}/\underline{50.67}$ | $\underline{21.12}/\underline{48.38}$ | $\underline{16.17}/\underline{40.41}$
$+$ _lvb_ and _ita_ monolingual FT with N3 | N5 | $21.49/49.94$ | $20.53/48.16$ | $15.10/39.47$
RBMT apertium-lld-ita | R1 | $11.38/39.72$ | $11.60/41.49$ | $8.48/34.48$
BM fine-tuned with authentic data |  |  |  |
$+$ _lvb_ monolingual BT with R1 | R2 | $14.43/42.76$ | $13.27/42.00$ | $13.99/37.37$
$+$ _ita_ monolingual BT with R2 | R3 | $22.17/50.33$ | $19.27/48.17$ | $15.89/40.19$
$+$ _lvb_ and _ita_ monolingual BT with R3 | R4 | $21.36/50.24$ | $20.27/\underline{49.08}$ | $16.34/\underline{\textbf{40.76}}$
$+$ _lvb_ and _ita_ monolingual FT with R3 | R5 | $\underline{22.50}/\underline{50.64}$ | $\underline{20.37}/49.04$ | $\underline{\textbf{16.36}}/40.47$
LLM gpt-3.5-turbo-0125 | L1 | $\underline{\textbf{26.77}}/\underline{\textbf{53.20}}$ | $21.17/48.52$ | $10.37/32.36$
BM fine-tuned with authentic data |  |  |  |
$+$ _lvb_ monolingual BT with L1 | L2 | $12.93/43.20$ | $12.21/41.21$ | $13.22/36.94$
$+$ _ita_ monolingual BT with L2 | L3 | $22.69/50.74$ | $20.37/48.40$ | $\underline{15.26}/38.99$
$+$ _lvb_ and _ita_ monolingual BT with L3 | L4 | $23.01/51.17$ | $\underline{21.38}/\underline{49.24}$ | $15.12/\underline{39.37}$
$+$ _lvb_ and _ita_ monolingual FT with L3 | L5 | $23.11/50.84$ | $20.86/48.50$ | $15.19/39.29$
ALL BM fine-tuned with authentic data |  |  |  |
$+$ _lvb_ and _ita_ monolingual BT with N3, R3, L3 | A1 | $23.58/50.68$ | $21.30/48.78$ | $15.32/39.56$
$+$ _lvb_ and _ita_ monolingual FT with N3, R3, L3 | A2 | $\underline{24.12}/\underline{51.42}$ | $\underline{\textbf{22.24}}/\underline{\textbf{49.69}}$ | $\underline{15.98}/\underline{39.64}$
Table 3:  Evaluation Results for Ladin to Italian Translation

##  5 Results and Discussion

The results of the various experiments conducted are presented in Table 3 and
Table 4, where the SacreBLEU Post (2018) and chrF++ Popović (2015) scores for
different models and test sets are detailed. To facilitate comparison, the
best scores for each approach have been underlined, and the overall best
scores for each testset are highlighted in bold.

Additionally, as recommended in Edunov et al. (2020), in Table 2 we report the
mean perplexity values for the Italian translations generated by the different
models to complement BLEU’s emphasis on adequacy. Perplexity measures how well
a language model can predict the next word in a sequence based on the
preceding words. Lower perplexity means that the model is more confident and
accurate in its predictions, indicating that it can better reproduce the
structure and patterns of the language it generates. Therefore, we present the
mean perplexity values obtained from GPT-2 Radford et al. (2019) , computed
using the implementation available from Hugging
Face171717https://huggingface.co/spaces/evaluate-metric/perplexity.

Several findings can be deduced from these results, and will be discussed
below. In general, there is evidence that augmenting the training data with
monolingual data through back-translation is effective. N1 shows that fine-
tuning the model with only authentic training data substantially improves the
results in both directions (compared to BM) in terms of BLEU/chrF++ points.
This shows on the one hand that the training is effective and on the other
hand that the available data is adequate. However, it is also evident that the
model generates less fluent text, as indicated by the perplexity scores which
increase for this model.

The results reveal a progression in difficulty among the test sets, where
Testset 3 emerges as the most challenging one. On this test set, all
approaches achieve similar low scores, suggesting the presented approach may
face limitations with more complex texts.

|  | Testset 1 | Testset 2 | Testset 3
---|---|---|---|---
Italian $\rightarrow$ Ladin (Val Badia) |  | BLEU / chrF++ | BLEU / chrF++ | BLEU / chrF++
NMT opus-mt-ine-ine | BM | $0.08/5.34$ | $0.55/13.68$ | $0.05/6.86$
BM fine-tuned with authentic data | N1 | $10.22/37.11$ | $10.14/37.48$ | $12.76/35.31$
$+$ _lvb_ monolingual BT with N1 | N2 | $19.09/46.92$ | $18.05/45.44$ | $16.50/37.46$
$+$ _ita_ monolingual BT with N2 | N3 | $19.54/\underline{47.02}$ | $\underline{19.45}/\underline{46.21}$ | $\underline{\textbf{16.66}}/37.36$
$+$ _lvb_ and _ita_ monolingual BT with N3 | N4 | $19.61/46.35$ | $19.16/45.63$ | $16.40/\underline{37.84}$
$+$ _lvb_ and _ita_ monolingual FT with N3 | N5 | $\underline{20.24}/46.72$ | $19.39/45.88$ | $15.56/36.97$
RBMT apertium-lld-ita | R1 | $4.94/37.50$ | $4.50/36.89$ | $3.19/27.44$
BM fine-tuned with authentic data |  |  |  |
$+$ _lvb_ monolingual BT with R1 | R2 | $19.18/46.59$ | $16.96/44.97$ | $15.21/36.76$
$+$ _ita_ monolingual BT with R2 | R3 | $19.86/46.83$ | $17.70/45.69$ | $15.04/36.60$
$+$ _lvb_ and _ita_ monolingual BT with R3 | R4 | $\underline{20.93}/\underline{47.65}$ | $\underline{19.32}/\underline{46.58}$ | $\underline{16.65}/\underline{\textbf{38.16}}$
$+$ _lvb_ and _ita_ monolingual FT with R3 | R5 | $19.97/46.88$ | $18.65/46.19$ | $16.61/38.12$
LLM gpt-3.5-turbo-0125 | L1 | $5.54/29.03$ | $3.84/28.98$ | $1.16/18.60$
BM fine-tuned with authentic data |  |  |  |
$+$ _lvb_ monolingual BT with L1 | L2 | $\underline{\textbf{22.09}}/\underline{\textbf{48.69}}$ | $19.71/46.59$ | $14.16/35.67$
$+$ _ita_ monolingual BT with L2 | L3 | $21.59/48.23$ | $\underline{\textbf{19.96}}/\underline{\textbf{49.96}}$ | $14.23/35.81$
$+$ _lvb_ and _ita_ monolingual BT with L3 | L4 | $20.82/47.86$ | $19.87/46.59$ | $\underline{16.55}/\underline{38.04}$
$+$ _lvb_ and _ita_ monolingual FT with L3 | L5 | $20.93/47.70$ | $19.38/46.37$ | $15.84/37.29$
ALL BM fine-tuned with authentic data |  |  |  |
$+$ _lvb_ and _ita_ monolingual BT with N3, R3, L3 | A1 | $19.83/47.16$ | $\underline{19.94}/\underline{46.40}$ | $\underline{16.54}/\underline{37.91}$
$+$ _lvb_ and _ita_ monolingual FT with N3, R3, L3 | A2 | $\underline{20.81}/\underline{47.50}$ | $19.71/46.36$ | $16.36/37.82$
Table 4:  Evaluation Results for Italian to Ladin Translation

In the translation direction _lvb_ $\rightarrow$ _ita_ , the best results were
achieved by combining the different back-translations, as model A2 results
indicate. This emphasises the importance of a broad and diversified dataset.
Remarkably, the A2 model is also competitive in the reverse translation
direction (_ita_ to _lvb_), although it does not achieve the best results.

A comparison of the models N1, R1 and L1 suggests that the LLM generates more
fluent texts (low perplexity) but perhaps does not always accurately reproduce
the meaning, as attested by the performance on Testset 3 (low perplexity but
also low BLEU score). In this assessment, the RBMT system R1 also performs
better than the fine-tuned NMT model N1. One of the reasons for the high
perplexity values of N1 is that this model tends to hallucinate because it has
been fine-tuned with a small data set. However, this does not seem to affect
the performance as the models trained on this data do not perform considerably
worse.

The performance of the LLM varies significantly, with pronounced differences
between the three test sets in both directions of translation. The significant
difference observed between Testset 1 and Testset 2 in the translation
direction from _lvb_ $\rightarrow$ _ita_ cannot be seen in the R\- and
N-models. It remains unclear to what extent the LLM benefits from the given
examples in the prompt. However, by providing an example, the propensity for
errors was minimised, resulting in fewer mistakes during execution. Even
though LLMs are not (yet) suitable for generating texts in low-resource
languages out-of-the-box (see performance of L1 in Table 4), Ladin and low-
resource languages in general could benefit from this technology. Our
experiments show that models trained on back-translations from L1 performed
best on Testset 1 and Testset 2 in the translation direction _ita_
$\rightarrow$ _lvb_.

The inclusion of forward translations in the training data did not
consistently improve the models, with the exception of the R-models for _lvb_
$\rightarrow$ _ita_. This suggests that these synthesised texts introduce too
much noise. However, model A2 was able to benefit from this data in the
translation direction _lvb_ $\rightarrow$ _ita_. Filtering this data could
slightly improve the model.

As the models achieve similar scores on the test data, we also examined the
quality of round-trip translations to gain additional insights. For this, we
used 10k sentences from the monolingual Ladin and Italian data (which were
also used for training, hence the high scores), translated them into the other
language and then back-translated them. This concept of so-called _round-trip
translation_ is a suitable evaluation method Zhuo et al. (2023). We used the
R4/N4/L4 models for this purpose, applying one model $A$ for one direction and
the same or a different model $B$ for the opposite direction. Table 5 shows
the obtained results. It can be clearly seen that the results are worse when a
different model is used for the reverse translation. This shows that although
the models achieve similar results with the test data, they work differently.
The R4 model proves to be the most stable here, as its translations can be
back-translated well by all three models. For other combinations, a high
variance can be observed.

The translation models N4181818https://doi.org/10.57967/HF/2695,
R4191919https://doi.org/10.57967/HF/2693, and
L4202020https://doi.org/10.57967/HF/2694 have been released on Hugging Face,
making them accessible for further research and application.

$A/B$ |  _lvb_ $\xrightarrow[]{A}$ _ita_ $\xrightarrow[]{B}$ _lvb_ |  _ita_ $\xrightarrow[]{A}$ _lvb_ $\xrightarrow[]{B}$ _ita_
---|---|---
| BLEU / chrF++ | BLEU / chrF++
N4/N4 |  70.57 / 82.56 |  64.19 / 81.26
N4/R4 | 58.57 / 74.50 | 47.16 / 72.09
N4/L4 | 63.90 / 78.09 | 59.47 / 78.46
R4/N4 | 70.80 / 82.20 | 68.38 / 83.00
R4/R4 |  80.12 / 88.94 |  68.51 / 84.73
R4/L4 | 70.36 / 81.98 | 67.41 / 82.68
L4/N4 | 63.72 / 77.53 | 57.02 / 76.54
L4/R4 | 57.13 / 73.32 | 46.95 / 71.52
L4/L4 |  72.31 / 83.69 |  65.74 / 82.02
Table 5:  Results for Round-Trip Translations

##  6 Related Work

Data augmentation such as back-translation Sennrich et al. (2016); Hoang et
al. (2018) and transfer learning Zoph et al. (2016) are established strategies
to improve MT systems. These concepts are discussed in Haddow et al. (2022);
Ranathunga et al. (2023), with a focus on low-resource scenarios. The fact
that the synthesised data plays a critical role in the quality of the systems
trained on it, as it also introduces a certain degree of noise, was discussed
extensively in Edunov et al. (2018); Xu et al. (2022). It was shown that
tagging synthetic data can be beneficial in the training process Caswell et
al. (2019). In our work we do not apply advanced techniques to differentiate
synthetic from real translations in training. The fact that RBMT systems can
still be valuable for low-resource languages and can even help to achieve
better results was also demonstrated for Northern Sámi Aulamo et al. (2021) .
In our experiments, we could also observe this in the translation direction
_lvb_ $\rightarrow$ _ita_. This could be due to the ability of the RBMT system
to provide general knowledge that is not available in the relatively limited
parallel training datasets Aulamo et al. (2021). The use of LLMs for MT and
different prompting techniques was investigated in Zhang et al. (2023) and
their performance in the machine translation of low-resource languages has
already been analysed in Moslem et al. (2023). Even if they struggle to
generate texts in low-resource languages Robinson et al. (2023), it has
already been claimed that they can contribute to advances in machine
translation of such languages. Our work is an example of how LLMs can be used
in machine translation of a low-resource language; however, further prompt
engineering is needed to make better use of such models.

##  7 Conclusion

In this work, we conducted a detailed comparison of RBMT, NMT and LLMs for
back-translation in a low-resource scenario. We have tested various back-
translation approaches and evaluated them for a previously unexplored language
in the field of machine translation.

Our current methodology involved the exclusion of numerous Ladin monolingual
sentences. However, this filtering would be less important for the translation
direction _lvb_ $\rightarrow$ _ita_. This previously discarded data could be
re-incorporated to improve the performance of the models in this particular
translation direction.

The round-trip translation scores indicate that the initial back-translation
with the RBMT system leads to more robust models. Improving the ambiguity
resolution of this rule-based translation system could lead to even better
results.

The simplicity of the prompts used to feed the LLMs provides a further
starting point for investigations. In particular, the question arises as to
whether the results can be improved by further prompt engineering, e.g., by
including the meaning for the distinct words occurring in a text using the
available dictionary. Investigating the effects of prompt optimisation could
provide new insights into maximising the efficiency of LLMs in machine
translation, especially in low-resource scenarios.

We plan to address these research questions in our future work.

## Acknowledgements

This research was conducted in collaboration with the Ladin Cultural Institute
_Micurà de Rü_ and funded by the _Regione Autonoma Trentino-Alto
Adige/Südtirol_.

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##  Appendix A Prompt template

⬇

I’ll give you samples for the translation from Ladin to Italian:

{

"translations": [

{

"Ladin": "scrí sües minunghes",

"Italian": "scrivere le proprie opinioni"

},

{

"Ladin": "mëte la secunda",

"Italian": "mettere la seconda"

},

{

"Ladin": ’"zessa, i á prescia!"’,

"Italian": ’"scansati, ho fretta!"’

},

{

"Ladin": "passé ia le rü",

"Italian": "oltrepassare il fiume"

},

...

{

"Ladin": "chësc liber é to",

"Italian": "questo libro è tuo"

},

]

}

Please generate the translation of each of the 16 entries in the given
dictionary, where the translations are empty. Return the same JSON dictionary
where the values for Italian are filled:

{

"translations": [

{

"Ladin": "Sperun da salvé almanco val’, dijun:",

"Italian": ""

},

{

"Ladin": "Ince tröc toponims y cognoms ladins desmostra che l’identité ladina
é coliada ala natöra y ala cultura da munt",

"Italian": ""

},

{

"Ladin": "De profesciun este pech... co este pa rové pro chësc laur?",

"Italian": ""

},

...

{

"Ladin": "I dormi n pü’ domisdé y spo ciamó val’ ora dan mesanöt",

"Italian": ""

}

]

}

Listing 1: Prompt template used to obtain the Italian translations from the
LLM.


# On the length of an arithmetic progression of the form ${3^{x}+2^{y}}$

Hongnan Chen<EMAIL_ADDRESS>College of Science, University of Shanghai for
Science and Technology, Shanghai
Shanghai, 200093, China
Fenglin Huang  School of Mathematical and Sciences, Peking University, Beijing
Beijing, 100091, China
<EMAIL_ADDRESS>Sihui Zhang111 corresponding author College of
Science, University of Shanghai for Science and Technology, Shanghai
Shanghai, 200093, China
<EMAIL_ADDRESS>
###### Abstract

The conclusion that the length of an arithmetic progression of the form
${3^{x}+2^{y}}$ is at most six is proved.

###### keywords:

arithmetic progression; representation.

Mathematics Subject Classification 2020: 11D85, 11B25

##  1 Introduction

The problem of a positive integers are represented as ${3^{x}+2^{y}}$ has been
an interestng topic in the theoretical study of mathematics. Iannucci, D.E
proved in [1] that there are exactly five positive integers that can be
written in more than one way as the sum of a nonnegative power of 2 and a
nonnegative power of 3. This proposition is given below.

######  Proposition 1.1.

(cf. [1]) There are exactly five positive integers that can be written in
multiple ways as the sum of non-negative powers of 2 and non-negative powers
of 3, and there are only five integers that can be expressed in two ways as
${3^{a}+2^{b}\left(a,b\in N\right)}$: 5, 11, 17, 35, 259. The five elements
listed are

$5=2^{2}+3^{0}=2^{1}+3^{1},\ 11=2^{3}+3^{1}=2^{1}+3^{2},\ \
17=2^{4}+3^{0}=2^{3}+3^{2},$ $35=2^{5}+3^{1}=2^{3}+3^{3},\ \
259=2^{8}+3^{1}=2^{4}+3^{5}.$

Pillai states that for each fixed integer $c\geq 1$, the Diophantine equation

$a^{x}-b^{y}=c,min{\left\\{x,y\right\\}}\geq 2$

has only a finite number of positive solutions ${a,b,x,y}$. This problem is
still open, it is still attracting a large number of researchers to study this
issue.

M. Ddamulira in [2] found that all integers $c$ having at least two
representations as a difference between a Fibonacci number and a power of 3.
Later, M. Ddamulira et al in [3] porved that all integers $c$ with at least
two representations as a difference between a k-generalized Fibonacci number
and a power of 3. In this paper, we prove the following conclusion.

######  Theorem 1.2.

The length of an arithmetic progression of the form ${3^{x}+2^{y}}$ for
nonnegative integers $x$ and $y$ is at most six.

##  2 Proof of Theorem 1.2

The example of an arithmetic progression of the form ${3^{x}+2^{y}}$ of length
six for non-negative integers $x$ and $y$ is given below. The six elements
listed are

$3=2^{1}+3^{0},\ \ 5=2^{2}+3^{0}=2^{1}+3^{1},\ \ 7=2^{2}+3^{1},$
$9=2^{0}+3^{2},\ \ 11=2^{1}+3^{2}=2^{3}+3^{1},\ \ 13=2^{2}+3^{2}.$

In the following, we will prove that an arithmetic progression of the form
${3^{x}+2^{y}}$ of length seven for non-negative integers $x$ and $y$ does not
exist.

If there is an A.P. with a length of 7 in
$\left\\{3^{m}+2^{n}|m,n\in\mathbb{N}\right\\}$, let it to

${a_{1}=a+6d},{a_{2}=a+5d},\dots,{a_{6}=a+d},{a_{7}=a}.$

Let $n=\left[log_{2}a_{1}\right]$, $m=\left[log_{3}a_{1}\right]$.

When $m\leq 8$, we give the following program to enumerate, see appendix for
program details. Therefore, we consider the case $m\geq 9$.

###  2.1 The properties of $d$

In this section, we will prove the three properties that $d\geq 500$, $d$ is
even, and $3\mid d$. These three lemmas are given below.

######  Lemma 2.1.

$d\geq 500$.

######  Proof 2.2.

If $d<500$, since $m\geq 9$,

${a_{1}}\geq{3^{m}}\geq{3^{9}}=19683.$

Thus

${a_{7}}={{a_{1}}-6d}\geq{{a_{1}}-3000}\geq{\frac{5}{6}{a_{1}}}\ \
(with~{}\frac{{a_{1}}}{6}>3000).$

Thus ${a_{1}},{a_{2}},{a_{3}},{a_{4}},{a_{5}},{a_{6}},{a_{7}}$ is
${3^{m}+2^{y}}$ or ${2^{n}+3^{x}}$ type, this means that four of the numbers
are of the same type.

For a prime $p$ and a natural number $n$, we write $p||n$ if $p\mid n$ but
$p^{2}\nmid n$. We denote the p-valuation of $n$ by ${\nu_{p}}(n)$: i.e.,
${\nu_{p}}(n)=k$ if ${p^{k}}||n.$

(1) If we have four ${3^{m}+2^{y}}$, let ${3^{m}+2^{p_{1}}}$,
${3^{m}+2^{p_{2}}}$, ${3^{m}+2^{p_{3}}}$, ${3^{m}+2^{p_{4}}}$ with
${p_{1}}>{p_{2}}>{p_{3}}>{p_{4}}$. Since

${\nu_{2}}({2^{p_{1}}-2^{p_{2}}})\geq
2+{\nu_{2}}({2^{p_{3}}-2^{p_{4}}})\geq{2+{\nu_{2}}(d)},$

and

${\nu_{2}}({2^{p_{2}}-2^{p_{3}}})\geq
1+{\nu_{2}}({2^{p_{3}}-2^{p_{4}}})\geq{1+{\nu_{2}}(d)}.$

Thus

$\frac{{2^{p_{1}}-2^{p_{2}}}}{d}\geq 4,\frac{{2^{p_{2}}-2^{p_{3}}}}{d}\geq 2.$

Therefore

$(3^{m}+2^{p_{1}})-(3^{m}+2^{p_{4}})\geq 7d.$

This is a contradiction.

(2) If we have four ${3^{x}+2^{n}}$, let ${3^{q_{1}}+2^{n}}$,
${3^{q_{2}}+2^{n}}$, ${3^{q_{3}}+2^{n}}$, ${3^{q_{4}}+2^{n}}$ with
${q_{1}}>{q_{2}}>{q_{3}}>{q_{4}}$. Since

${\nu_{3}}({3^{q_{1}}-3^{q_{2}}})\geq
2+{\nu_{2}}({3^{q_{3}}-3^{q_{4}}})\geq{2+{\nu_{3}}(d)},$

thus

$\frac{{3^{q_{1}}-3^{q_{2}}}}{d}\geq 9,$

therefore

$(3^{q_{1}}+2^{n})-(3^{q_{2}}+2^{n})\geq 9d.$

This is a contradiction.

######  Lemma 2.3.

$2\mid d$.

######  Proof 2.4.

If d is odd, then there must exist three even numbers and all are denoted as
$3^{x}+2^{y}$. Since $2\mid 3^{x}+2^{y}$, $3^{x}$ is odd, thus $y=0$.
Therefore, all three numbers are expressed in the form $3^{x}+1$. Thus, let
${1+3^{S_{1}}},{1+3^{S_{2}}},{1+3^{S_{3}}}$ where ${S_{1}}>{S_{2}}>{S_{3}}$.

Since ${S_{1}}>{S_{2}}$, thus

${1+3^{S_{1}}}-({1+3^{S_{2}}})={3^{S_{1}}-3^{S_{2}}}>3^{S_{2}}>{3^{S_{2}}-3^{S_{3}}}.$

This is a contradiction.

######  Remark 2.5.

If there exists ${y_{i}}=0$, then there must exist three even numbers. The
situation is the same as the first proof, this is a contradiction.

######  Lemma 2.6.

$3\mid d$.

######  Proof 2.7.

If $3\nmid d$, thus we consider that ${a_{i}}=3^{x_{i}}+2^{y_{i}}$,
${a_{i+1}}=3^{x_{i+1}}+2^{y_{i+1}}$, ${a_{i+2}}=3^{x_{i+2}}+2^{y_{i+2}}$.

Thus $d={a_{i}}-{a_{i+1}}={a_{i+1}}-{a_{i+2}}$ not a multiple of 3, and
$2d={a_{i}}-{a_{i+2}}$ not a multiple of 3.

We consider the parity of ${y_{i}}$, ${y_{i+1}}$, ${y_{i+2}}$, where two
numbers must have the same parity, thus their difference in powers of 2 must
be a multiple of 3, it may be useful to set the subscripts to j and k, thus
$3\mid(2^{y_{j}}-2^{y_{k}})$.

However $3\nmid d$, thus

${a_{j}}-{a_{k}}=(3^{x_{j}}+2^{y_{j}})-(3^{x_{k}}+2^{y_{k}})$

not a multiple of 3, thus $3^{x_{j}}-3^{x_{k}}$ not a multiple of 3.

This shows that ${x_{j}}=0$ or ${x_{k}}=0$, thereby at least one of the three
numbers ${x_{i}},{x_{i+1}},{x_{i+2}}$ is equal to 0, and $i=1,2,3,4,5$. Thus
at least one of the three numbers ${x_{1}},{x_{2}},{x_{3}}$ is equal to 0, and
at least one of the three numbers ${x_{4}},{x_{5}},{x_{6}}$ is equal to 0.

Thus at least two of the six numbers
${x_{1}},{x_{2}},{x_{3}},{x_{4}},{x_{5}},{x_{6}}$ are equal to 0, let $i,j$
such that ${x_{i}}={x_{j}}=0$ and $1\leq i<j\leq 6$, $(i,j)$ is the smallest
set of them. Then there must be

$j-i\leq 3$ (1)

otherwise $j-i>3$, there must be $0$ in the three numbers
${x_{i+1}},{x_{i+2}},{x_{i+3}}$, which contradicts the minimality of $i,j$.

Let ${a_{i}}=2^{y_{i}}+1,{a_{j}}=2^{y_{j}}+1$, since

${a_{j}}\geq{a_{6}}\geq{{a_{7}}+d},{a_{7}}\geq{3^{0}}+{2^{0}}=2,$

thus ${a_{j}}\geq d+2$. Meanwhile from (1) we have ${a_{i}}-{a_{j}}=(j-i)d\leq
3d$, thus

$2^{y_{i}}-2^{y_{j}}\leq 3d\leq 3({a_{j}}-2)=3\cdot(2^{y_{j}}+1-2)=3\cdot
2^{y_{j}}-3,$

thus ${2^{y_{i}}}\leq{4\cdot 2^{y_{j}}-3}$, therefore ${y_{i}}<{y_{j}}+2$.
Since ${y_{i}}>{y_{j}}$, we have ${y_{i}}={y_{j}}+1$, thus
${a_{i}}-{a_{j}}=2^{y_{j}}=(j-i)d$. Since $1\leq j-i\leq 3$, thus

$d=2^{y_{j}}~{}or~{}d=2^{{y_{j}}-1},$

thereby

$d\geq 2^{{y_{j}}-1},~{}j-i=1~{}or~{}2$ (2)

thus $j-i\neq 3$, therefore $j-i\leq 2$. Since $i\leq 3$, thus $j\leq 5$. This
implies that ${a_{j}}-2d\geq{a_{7}}$. From (2) we have

${a_{j}}-2d\leq(2^{y_{j}}+1)-2\cdot 2^{{y_{j}}-1}=1,$

thus ${a_{7}}\leq 1$. This is a contradiction.

###  2.2 The type of ${a_{1}},\dots,{a_{4}}$

In this section we discuss all the possible types of ${a_{1}},\dots,{a_{4}}$.
We show that there are four possible types for ${a_{4}}$, two possible types
for ${a_{1}}$ and ${a_{2}}$, and three possible types for ${a_{3}}$.

######  Proposition 2.8.

${a_{4}}=a+3d$ can only be written as one of the four numbers

${3^{m}+2^{y_{4}}},{3^{m-1}+2^{y_{4}}},{2^{n-1}+3^{x_{4}}},{2^{n}+3^{x_{4}}}.$

######  Proof 2.9.

$3^{m}\leq{a_{1}}=a+6d<2^{n+1},2^{n}\leq{a_{1}}=a+6d<3^{m+1}.$

Thus

$\displaystyle 3^{m-2}+2^{n-2}$ $\displaystyle=\frac{1}{9}\cdot
3^{m}+\frac{1}{4}\cdot 2^{n}<\frac{1}{9}\cdot 2^{n+1}+\frac{1}{4}\cdot 2^{n}$
$\displaystyle=\frac{17}{36}\cdot 2^{n}\leq\frac{17}{36}(a+6d)<a+3d.$

Thus $a+3d=3^{x_{4}}+2^{y_{4}}$ where ${x_{4}}\geq m-1$ or ${y_{4}}\geq n-1$.
Thus $a+3d$ has four possibilities:

${3^{m}+2^{y_{4}}},{3^{m-1}+2^{y_{4}}},{2^{n-1}+3^{x_{4}}},{2^{n}+3^{x_{4}}}.$

######  Proposition 2.10.

${a_{1}}=a+6d,{a_{2}}=a+5d$ must be expressed as

${3^{m}+2^{y}}~{}or~{}{2^{n}+3^{x}}.$

######  Proof 2.11.

If ${x_{2}}\neq m$ and ${y_{2}}\neq n$. Since ${x_{2}}\leq m,{y_{2}}\leq n$,
thus

${x_{2}}\leq m-1,{y_{2}}\leq n-1.$

Therefore

$\displaystyle a+5d$ $\displaystyle=3^{x_{2}}+2^{y_{2}}\leq
3^{m-1}+2^{n-1}=\frac{1}{3}\cdot 3^{m}+\frac{1}{2}\cdot 2^{n}$
$\displaystyle\leq\frac{1}{3}(a+6d)+\frac{1}{2}(a+6d)=\frac{5}{6}a+5d.$

The above inequality holds because we have $2^{n}\leq a+6d,3^{m}\leq a+6d$.
This is a contradiction.

Thus ${x_{2}}=m$ or ${y_{2}}=n$, thereby ${a_{2}}=a+5d={3^{m}+2^{y_{2}}}$ or
${2^{n}+3^{x_{2}}}$.

By the same token we can obtain ${x_{1}}=m$ or ${y_{1}}=n$, thereby
${a_{1}}=a+6d={3^{m}+2^{y_{1}}}$ or ${2^{n}+3^{x_{1}}}$.

######  Proposition 2.12.

${a_{3}}=a+4d$ can only be written as one of the three numbers

${3^{m}+2^{y}},{2^{n}+3^{x}},{3^{m-1}+2^{n-1}}.$

######  Proof 2.13.

we consider that

$\displaystyle 2^{n-1}+3^{m-2}$ $\displaystyle=\frac{1}{2}\cdot
2^{n}+\frac{1}{9}\cdot 3^{m}\leq\frac{1}{2}\cdot(a+6d)+\frac{1}{9}\cdot(a+6d)$
$\displaystyle=\frac{11}{18}\cdot a+\frac{11}{3}\cdot d<\frac{11}{18}\cdot
a+4d<a+4d,$

$\displaystyle 2^{n-2}+3^{m-1}$ $\displaystyle=\frac{1}{4}\cdot
2^{n}+\frac{1}{3}\cdot 3^{m}\leq\frac{1}{4}\cdot(a+6d)+\frac{1}{3}\cdot(a+6d)$
$\displaystyle=\frac{7}{12}\cdot a+\frac{7}{2}\cdot d<\frac{7}{12}\cdot
a+4d<a+4d.$

Thus, $a+4d\geq 2^{n-1}+3^{m-2}$, and $a+4d\geq 2^{n-2}+3^{m-1}$, then
${a_{3}}=a+4d$ can only be written as one of the three numbers

${3^{m}+2^{y}},{2^{n}+3^{x}},{3^{m-1}+2^{n-1}}.$

###  2.3 ${a_{1}},\dots,{a_{4}}$ Specific discussion

In this section, for the four numbers ${3^{m}+2^{y_{i}}}$,
${3^{x_{i}}+2^{n}}$, ${3^{m-1}+2^{y_{i}}}$ and ${3^{x_{i}}+2^{n-1}}$, we will
call them $3$-dominated, $2$-dominated, $3$-weak dominated and $2$-weak
dominated, respectively.

Below we first prove that there cannot be three $3$-dominated or three
$2$-dominated or two $3$-dominated and two $2$-dominated in
${a_{1}},\dots,{a_{4}}$. Then we prove that ${a_{3}}\neq 3^{m-1}+2^{n-1}$,
${a_{4}}\neq{2^{n-1}+3^{x_{4}}}$ and ${a_{4}}\neq{3^{m-1}+2^{y_{4}}}$. In
summary, it can be shown that all cases of ${a_{1}},\dots,{a_{4}}$ are
contradictory.

######  Proposition 1.

There can be no three numbers in ${a_{1}},\dots,{a_{4}}$ expressed as
$3$-dominated.

######  Proof 2.14.

If there are three $3$-dominated in ${a_{1}},\dots,{a_{4}}$, denoted
${3^{m}+2^{p}},{3^{m}+2^{q}}$, ${3^{m}+2^{r}}$ where $p>q>r$. Then,

$d\mid({3^{m}+2^{p}})-({3^{m}+2^{q}})\leq 2d,$
$d\mid({3^{m}+2^{q}})-({3^{m}+2^{r}})\geq d,$

thus

${2^{p}-2^{q}}>{2^{q}-2^{r}}\geq d,$
${2^{p}-2^{q}}>{2^{q}-2^{r}}=({3^{m}+2^{q}})-({3^{m}+2^{r}})\equiv
0~{}(\mod~{}\mathrm{d}).$

Therefore

$d=({3^{m}+2^{q}})-({3^{m}+2^{r}})={2^{q}-2^{r}},$
$2d=({3^{m}+2^{p}})-({3^{m}+2^{q}})={2^{p}-2^{q}}={2^{q+1}-2^{r+1}}.$

From this, we can see

$3^{m}+2^{p}=a_{1},3^{m}+2^{q}=a_{3},3^{m}+2^{r}=a_{4}.$

Thus

$p=q+1=r+2,d={2^{q}-2^{r}}={2^{r+1}-2^{r}}=2^{r}.$

If ${a_{5}}=3^{x_{5}}+2^{y_{5}}={a_{4}}-d={({3^{m}+2^{r}})-2^{r}}=3^{m}\equiv
0\left(\mod~{}\mathrm{3}\right)$, thus ${x_{5}}=0$. Thus

$3^{x_{5}}+2^{y_{5}}=3^{0}+2^{y_{5}}=1+2^{y_{5}}=3^{m},$

thus

$2^{y_{5}}+3^{1}=3^{m}+2^{1}$ (3)

According to proposition 1.1, (3) is no solution if $m\geq 6$. This is a
contradiction.

######  Proposition 2.15.

There can be no three numbers in ${a_{1}},\dots,{a_{4}}$ expressed as
$2$-dominated.

######  Proof 2.16.

If there are three $2$-dominated in ${a_{1}},\dots,{a_{4}}$, denoted
${2^{n}+3^{p}}$, ${2^{n}+3^{q}}$, ${2^{n}+3^{r}}$ where $p>q>r$, thus

$3d\geq(2^{n}+3^{p})-(2^{n}+3^{r})=3^{p}-3^{r}\geq
3^{q+1}-3^{r}>3(3^{q}-3^{r})\geq 3d.$

This is a contradiction.

######  Proposition 2.17.

There cannot be represented as exactly two $2$-dominated and two
$3$-dominated.

######  Proof 2.18.

If ${a_{1}},\dots,{a_{4}}$ is the arrangement of
${2^{n}+3^{x}},{2^{n}+3^{y}},{3^{m}+2^{z}}$ and ${3^{m}+2^{w}}$, without loss
of generality, we assume $x>y$ and $z>w$.

(1) If ${a_{1}},{a_{2}}$ are the same type, then ${a_{3}},{a_{4}}$ are the
same type, thus

$d={3^{x}-3^{y}}={2^{z}-2^{w}},$

thereby $d<500.$ This is a contradiction.

(2) If ${a_{1}},{a_{3}}$ are the same type, then ${a_{2}},{a_{4}}$ are the
same type. thus

$2d={3^{x}-3^{y}}={2^{z}-2^{w}},$

thereby $d<500.$ This is a contradiction.

(3) If ${a_{1}}={2^{n}+3^{x}},{a_{4}}={2^{n}+3^{y}}$. Thus
${3d={a_{1}}-{a_{4}}=3^{x}-3^{y}}$ where $y>1$, and

${a_{2}}-{a_{3}}=d=({3^{m}+2^{z}})-({3^{m}+2^{w}})={2^{z}}-{2^{w}}=3^{x-1}-3^{y-1},$

thereby $d<500$. This is a contradiction.

(4) If ${a_{1}}={3^{m}+2^{z}},{a_{4}}={3^{m}+2^{w}}$. Thus
${3d={a_{1}}-{a_{4}}=2^{z}-2^{w}}$, and ${d={a_{2}}-{a_{3}}=3^{x}-3^{y}}$.
Thus

$3d=2^{z}-2^{w}=3^{x+1}-3^{y+1},$

thereby $d<500$. This is a contradiction.

Combining Proposition 1, 2.15 and 2.17, ${a_{1}},\dots,{a_{4}}$ can’t be all
$3$-dominated or $2$-dominated.

######  Proposition 2.19.

${a_{3}}\neq 3^{m-1}+2^{n-1}$.

######  Proof 2.20.

According to a lemma 2.1, 2.3, 2.6, we obtain that: ${6\mid d}$, $d\geq 500$,
$y_{i}$ is a positive integer for all $a+(7-i)d=3^{x_{i}}+2^{y_{i}}$.

It has already been demonstrated that ${a_{1}},{a_{2}}$ is $3$-dominated or
$2$-dominated type, ${a_{3}}$ can only be $3$-dominated, $2$-dominated,
${3^{m-1}+2^{n-1}}$ and ${a_{4}}$ can only be $3$-dominated, $2$-dominated,
$3$-weak dominated, $2$-weak dominated.

If ${a_{3}}=3^{m-1}+2^{n-1}$, the possibility of ${a_{1}}$ is discussed below.

(1) If ${a_{1}}=3^{m}+2^{y_{1}}$, we have

$4\mid 2d={a_{1}}-{a_{3}}=2\cdot 3^{m-1}+2^{y_{1}}-2^{n-1}.$

Thus ${y_{1}}=1,{a_{1}}=3^{m}+2$. Applying Proposition 2.10 and the fact that
$6\mid d$, we obtain that ${a_{2}}=3^{x_{2}}+2^{n}$, thus

$3\mid d={a_{2}}-{a_{3}}=(3^{x_{2}}+2^{n})-(3^{m-1}+2^{n-1}),$

therefore ${x_{2}}=0$, ${a_{2}}=1+2^{n}$, thus
${a_{1}}+{a_{3}}=2\cdot{a_{2}}$, thereby $3^{m}+2+3^{m-1}+2^{n-1}=2\cdot
2^{n+1}$, therefore $4\cdot 3^{m-1}=3\cdot 2^{n-1}$, thus

$3^{m-2}=2^{n-3}.$

Combing with the fact that $m\geq 6$ yields a contradiction.

(2) If ${a_{1}}=3^{x_{1}}+2^{n}$, thus

$3\mid 2d={a_{1}}-{a_{3}}=3^{x_{1}}+2^{n}-2^{n-1}-3^{m-1},$

thereby ${x_{1}}=0$, thus ${a_{1}}=2^{n}+1$. Applying Proposition 2.10 shows
that

${a_{2}}=3^{m}+2^{y_{2}},$ $4\mid
2d={a_{1}}-{a_{3}}=2^{n}+1-2^{n-1}-3^{m-1}=2^{n-1}-(3^{m-1}-1).$

Thus $4\mid 3^{m-1}-1$, $2\mid m-1$, $8\mid 3^{m-1}-1$, therefore $8\mid 2d$,
thus

$4\mid d={a_{2}}-{a_{3}}=3^{m}+2^{y_{2}}-2^{n-1}-3^{m-1}=2\cdot
3^{m-1}+2^{y_{2}}-2^{n-1}.$

Thus ${y_{2}}=1$, thereby ${a_{2}}=3^{m}+2$, and
${a_{1}}+{a_{3}}=2\cdot{a_{2}}$, thus

$(2^{n}+1)+(3^{m-1}+2^{n-1})=2\cdot 3^{m}+4.$

Thus $5\cdot 3^{m-1}=3\cdot 2^{n-1}-3$, thus $5\cdot 3^{m-2}=2^{n-1}-1$. Since
$m\geq 6$, thus $9\mid 2^{n-1}-1$, thereby $6\mid n-1$, thereby $63\mid
2^{n-1}-1$.

Finally, it can be introduced that $7\mid 2^{n-1}-1$. But $7\nmid 5\cdot
3^{m-2},$ this is a contradiction.

In conclusion, ${a_{3}}\neq 3^{m-1}+2^{n-1}$, thus ${a_{1}},{a_{2}},{a_{3}}$
is $3$-dominated or $2$-dominated, therefore ${a_{4}}$ is not this type.
Applying Proposition 2.8, it suffices to consider the following two scenarios:
${a_{4}}={3^{m-1}+2^{y_{4}}}$ or ${2^{n-1}+3^{x_{4}}}$.

######  Proposition 2.21.

${a_{4}}\neq{2^{n-1}+3^{x_{4}}}$.

######  Proof 2.22.

When ${a_{4}}=3^{x_{4}}+2^{n-1},$ then

$2^{n}+2\cdot 3^{x_{4}}=2\cdot{a_{4}}={a_{1}}+{a_{7}}>2^{n}+2.$

where the inequality comes from the fact that Lemma 2.3,
${a_{1}},\dots,{a_{7}}$ is odd and ${a_{1}}\geq 2^{n}$. Thus ${a_{1}}\geq
2^{n}+1$, ${a_{7}}\geq 2^{1}+1$, thus ${x_{4}}\geq 1$.

Since ${a_{1}},{a_{2}},{a_{3}},{a_{4}}$ not three $3$-dominated, there must be
${2^{n}+3^{x}}$ in ${a_{1}},{a_{2}},{a_{3}}$, take the one with the smallest
$x$. Since

$3\mid d\mid 2^{n}+3^{x}-(3^{x_{4}}+2^{n-1}),$

thereby $x=0~{}({x_{4}}\geq 0)$. Thus there exists $1\leq i\leq 3$,
${a_{i}}=2^{n}+1$.

For $1\leq j\leq 3$, $j\neq i$,

$3\mid d\mid({a_{j}}-{a_{i}})=3^{x_{j}}+(2^{y_{j}}-2^{n})-1.$

If ${x_{j}}=0$, thus ${y_{j}}=n$, ${a_{i}}={a_{j}}$, this is a contradiction.
Thus ${x_{j}}\geq 1$, $3\mid 2^{y_{j}}-2^{n}$. Therefore, ${x_{j}}=m$, thus

${a_{j}}=3^{m}+2^{y_{j}}.$

We consider the three cases of

${a_{i}}=2^{n}+1,1\leq i\leq 3.$

Case 1: ${a_{3}}=2^{n}+1$. Then

${a_{1}}={3^{m}+2^{y_{1}}},{a_{2}}=3^{m}+2^{y_{2}},{a_{3}}={1+2^{n}},{a_{4}}={3^{x_{4}}+2^{n-1}}.$

Since ${3^{m}\leq a+6d\leq 2^{n+1}}$, thus

$\displaystyle{a+3d}={a_{4}}$
$\displaystyle={3^{x_{4}}+2^{n-1}}\leq{3^{m-2}+2^{n-1}}$
$\displaystyle\leq\frac{1}{2}\cdot{2^{n}}+\frac{1}{9}\cdot{3^{m}}\leq\frac{1}{2}\cdot{2^{n}}+\frac{1}{9}\cdot{2^{n+1}}$
$\displaystyle=\frac{13}{18}\cdot{2^{n}}<\frac{3}{4}\cdot{2^{n}}<\frac{3}{4}(2^{n}+1)=\frac{3}{4}(a+4d).$

This is a contradiction.

Case 2: ${a_{2}}=2^{n}+1$. Then

${a_{1}}={3^{m}+2^{y_{1}}},{a_{2}}={1+2^{n}},{a_{3}}={3^{m}+2^{y_{3}}},{a_{4}}={3^{x_{4}}+2^{n-1}}.$

Since ${4(a+3d)>3(a+4d)}$, thereby $4(3^{m-2}+2^{n-1})>3\cdot 3^{m}$, thereby

${2^{n+1}>23\cdot 3^{m-2}}.$

Meanwhile ${5(a+3d)>3(a+4d)}$, thereby $5(3^{m-2}+2^{n-1})>3\cdot 2^{n}$,
thereby

${5\cdot 3^{m-2}>2^{n-1}}.$

From above we obtain

$2^{n-1}>\frac{23}{4}\cdot 3^{m-2}>{5\cdot 3^{m-2}>2^{n-1}},$

this is a contradiction.

Case 3: ${a_{1}}=2^{n}+1$. Then

${a_{1}}={1+2^{n}},{a_{2}}={3^{m}+2^{y_{2}}},{a_{3}}={3^{m}+2^{y_{3}}},{a_{4}}={3^{x_{4}}+2^{n-1}}.$

(1) If ${\nu_{2}}(d)=1.$

Since ${y_{2}}>{y_{3}}\geq 1$, thereby ${y_{2}}\geq 2$; Since ${2\mid d}$,
thereby

$4\mid 2d=(3^{m}+2^{y_{2}})-(3^{x_{4}}+2^{n-1})\equiv
3^{m}-3^{x_{4}}~{}(\mod~{}4).$

Therefore, ${x_{4}}$ has the same parity as $m$.

Since ${\nu_{2}}(d)=1$, we have

$4\nmid{3^{m}+2^{y_{3}}-3^{x_{4}}-2^{n-1}},$

thereby ${y_{3}}=1$. But ${\nu_{2}}(d)=1$, thus ${\nu_{2}}(2d)=2.$ Thereby

${{\nu_{2}}(3^{m}+2^{y_{2}}-3^{x_{4}}-2^{n-1})=2},$

thus ${y_{2}}=2$, which implies that $d=2$. This is a contradiction.

(2) If ${\nu_{2}}(d)\geq 2$.

we hace

$4\mid 3d=({2^{n}}+1)(2^{n-1}+3^{x_{4}}).$

Therefore, ${x_{4}}$ is an even number, thus

$3d\equiv{3^{x_{4}}-1}\equiv 0~{}(\mod~{}8),$

thereby $8\mid d$.

If $y_{2}\leq n-5$, then combining with the fact that ${5(a+6d)}<{6(a+5d)}$
and ${3(a+4d)}<{4(a+3d)}$, it yields that ${77\cdot 2^{n-4}<6\cdot 3^{m}}$ and
${2^{n+1}>23\cdot 3^{m-2}}$. This is a contradiction. Thus ${y_{2}}\geq n-4$.

Since ${m>6}$, we get that ${n>7}$ and ${{y_{2}}\geq 4}$. Thus

${16\mid
2d}={2^{y_{2}}+3^{m}-(2^{n-1}+3^{x_{4}})\equiv{3^{m}-3^{x_{4}}}(\mod~{}16)},$

thereby ${x_{4}}\equiv m~{}(\mod~{}4)$, thus ${x_{4}}\leq m-4$.

Since

${3(a+4d)<4(a+3d)},$

thus

${2^{n+1}>239\cdot 3^{m-4}};$ (4)

If ${y_{2}}\leq n-3$, since

${5(a+6d)<6(a+5d)},$

thus

$5(2^{n}+1)<6(3^{m}+2^{n+3}),$

thereby

$7\cdot 2^{n-2}<6\cdot 3^{m}.$ (5)

Combining (4) and (5) shows that this is contradictory.

Therefore ${y_{2}}>n-3$, and

$3\mid d=2^{n}+1-(3^{m}+2^{y_{2}}),$

therefore, ${y_{2}}$ and $n$ have different parity, thus ${y_{2}}=n-1$.

Since

$5(a+3d)>3(a+5d),$

thus

$5(3^{x_{4}}+2^{n-1})>3(3^{m}+2^{n-1}),$

thereby

${2^{n}}>{3^{m+1}-5\cdot 3^{x_{4}}}\geq{3^{m+1}-5\cdot 3^{m-4}}=238\cdot
3^{m-4}.$ (6)

Combining (4) and (6) shows that this is contradictory.

######  Proposition 2.23.

${a_{4}}\neq{3^{m-1}+2^{y_{4}}}$.

######  Proof 2.24.

If ${a_{4}}=3^{m-1}+2^{y_{4}}$. Since

${2\cdot 3^{m-1}+2^{{y_{4}}+1}=2{a_{4}}={a_{7}}+{a_{1}}>3^{m}},$

and $m>11$, thus ${y_{4}}>4$.

Since there must be a number in ${a_{1}},{a_{2}},{a_{3}}$ that is
$3$-dominated, suppose $3^{m}+2^{y}$ is smallest. Since $d$ is a multiple of
2, therefore

$2\mid 3^{m}+2^{y}-(3^{m-1}+2^{y_{4}}),$

Thus $y\geq 1$.

We are going to prove $y=1$ by contradiction.

If $y\geq 2$, thus

${3^{m}+2^{y}}-(3^{m-1}+2^{y_{4}})=2\cdot 3^{m-1}+2^{y}-2^{y_{4}}\equiv
2~{}(\mod~{}4).$

Thus $d$ is not a multiple of 4, thereby $d\equiv 2~{}(\mod~{}4)$. Thus

$2d={a_{2}}-{a_{4}}=3^{x_{2}}+2^{y_{2}}-(3^{m-1}+2^{y_{4}})\equiv
4~{}(\mod~{}8).$

This is discussed below in two cases.

Case 1: When ${x_{2}}\equiv m-1~{}(\mod~{}2)$, thus ${x_{2}}\neq m$, thus
${y_{2}}=n>3$, therefore

$2d\equiv 3^{x_{2}}-3^{m-1}\equiv 0~{}(\mod~{}8).$

This is a contradiction.

Case 2: When ${x_{2}}\not\equiv m-1~{}(\mod~{}2)$, thus ${3^{x_{2}}-3^{m-1}}$
not a multiple of 4, but $4\mid 2^{y_{4}}$, thus $2^{y_{2}}$ not a multiple of
4, therefore ${y_{2}}=1<y$. This contradicts the minimality of $y$.

Combining case 1 and case 2 shows that $y=1$. Thus there exists $1\leq i\leq
3$, such that

${a_{i}}=3^{m}+2.$

Since $d>1000$, therefore, for $1\leq j\leq 3$, $j\neq i$,
${a_{j}}\neq{3^{m}+4}$.

We consider that

${{a_{i}}-{a_{4}}}={3^{m}-3^{m-1}+2-2^{y_{4}}}={2\cdot(3^{m-1}+1)-2^{y_{4}}},$

If there exists $1\leq j\leq 3$ and $j\neq i$, such that
${a_{j}}=3^{m}+2^{y_{j}}$, thus ${y_{j}}>1$. Therefore

${a_{j}}-{a_{4}}=2\cdot 3^{m-1}+2^{y_{j}}-2^{y_{4}}\equiv 2~{}(\mod~{}4),$

thus $d\equiv 2~{}(\mod~{}4)$. Since ${a_{j}}-{a_{4}}$ is a multiple of 4,
thus $i=2,j=1$. Thus

${a_{1}}=3^{m}+2^{y_{1}},{a_{2}}=3^{m}+2,{a_{4}}=3^{m-1}+2^{y_{4}}.$

Applying Proposition 1, we have

${a_{3}}=2^{n}+3^{x_{3}}.$

From above we have

${a_{1}}=3^{m}+2^{y_{1}},{a_{2}}=3^{m}+2,{a_{3}}=2^{n}+3^{x_{3}},{a_{4}}=3^{m-1}+2^{y_{4}}.$

We consider that

$2d={a_{2}}-{a_{4}}=2\cdot(3^{m-1}+1)-2^{y_{4}}$

is not a multiple of 8. Thus $3^{m-1}\equiv 1~{}(\mod~{}8)$, therefore

${a_{4}}=1~{}(\mod~{}8),{a_{2}}=5~{}(\mod~{}8).$

Since $d>1000$, ${a_{1}}\geq{a_{2}}+1000$, thus ${y_{1}}\geq 4$, thereby

${a_{1}}\equiv 3^{m}\equiv 3~{}(\mod~{}8).$

Thus

${a_{3}}=3^{x_{3}}+2^{n}\equiv 7~{}(\mod~{}8),$

but

$3^{x_{3}}\equiv 1~{}(\mod~{}8)~{}or~{}3^{x_{3}}\equiv 3~{}(\mod~{}8).$

This is a contradiction.

Finally, we discuss each of the three cases of

${a_{i}}={3^{m}}+2,i=1,2,3.$

Case 1 : When ${a_{1}}={3^{m}}+2$, then

${a_{1}}={3^{m}+2},{a_{2}}={3^{x_{2}}+2^{n}},{a_{3}}={3^{x_{3}}+2^{n}},{a_{4}}={3^{m-1}+2^{y_{4}}}.$

Since

$\frac{3}{4}\cdot{a_{3}}=\frac{3}{4}(a+4d)\leq(a+3d)={a_{4}},$

and ${y_{4}}\leq n-2,$ thereby

$\frac{3}{4}(3^{x_{3}}+2^{n})\leq{3^{m-1}+2^{y_{4}}}\leq{3^{m-1}+2^{n-2}},$

thus

${2^{n-1}<3^{m-1}}.$

On the other hand, we have

${2(a+3d)=2{a_{4}}={a_{1}}+{a_{7}}\geq{a_{1}}={a+6d}},$

thereby

${(3^{m-1}+2^{y_{4}})\cdot 2\geq 3^{m}+2},$

thus

${2^{n-1}>3^{m-1}}.$

This is a contradiction.

Case 2 : When ${a_{2}}={3^{m}}+2$, then

${a_{1}}={3^{x_{1}}+2^{n}},{a_{2}}={3^{m}+2},{a_{3}}={3^{x_{3}}+2^{n}},{a_{4}}={3^{m-1}+2^{y_{4}}}.$

Since

$\frac{3}{4}\cdot{a_{3}}=\frac{3}{4}(a+4d)\leq(a+3d)={a_{4}},$

and ${y_{4}}\leq n-2.$ Therefore,

$\frac{3}{4}(3^{x_{3}}+2^{n})\leq{3^{m-1}+2^{y_{4}}},$

thus

${2^{n-1}<3^{m-1}}.$

On the other hand, since

${\frac{3}{5}(a+5d)<{a+3d}},$

thus

${3(3^{m}+2)<{5(3^{m-1}+2^{y_{4}})}\leq{5\cdot 3^{m-1}+5\cdot 2^{n-2}}},$

thus

${4\cdot 3^{m-1}<5\cdot 2^{n-2}},$

thereby

${3^{m-1}<2^{n-1}}.$

This is a contradiction.

Case 3 : When ${a_{3}}={3^{m}}+2$, then

${a_{1}}={3^{x_{1}}+2^{n}},{a_{2}}={3^{x_{2}}+2^{n}},{a_{3}}={3^{m}+2},{a_{4}}={3^{m-1}+2^{y_{4}}}.$

Since

$4(a+3d)>3(a+4d),$

thus

$4(3^{m-1}+2^{n-2})>3\cdot 3^{m},$

thereby

$2^{n}>5\cdot 3^{m-1}.$

On the other hand,

$5(a+3d)>3(a+5d),$

thus

$5(3^{m-1}+2^{n-2})>3\cdot 2^{n},$

thereby

$5\cdot 3^{m-1}>7\cdot 2^{n-2}>2^{n}.$

This is a contradiction.

From the proof of the above proposition, we obtain that there is no an
arithmetic progression of the form ${3^{x}+2^{y}}$ of lenth seven. The proof
of theorem 1.2 is complete.

##  Appendix A Appendices

Figure 1: program

## References

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  * [2] Ddamulira, M. (2020). On a problem of pillai with fibonacci numbers and powers of $3$. Boletín de la Sociedad Matemática Mexicana, 26(2), 263-277.
  * [3] Ddamulira, M. and Luca, F. (2020). On the problem of Pillai with k-generalized Fibonacci numbers and powers of $3$. International Journal of Number Theory, 16(07), 1643-1666.