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Journals / Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi / 2026 / Cilt: 23 Sayı: 4

Mulatu Numbers That Are Concatenations of Two Lucas Numbers

İki Lucas Sayısının Birleşimi Olan Mulatu Sayıları

Fatih Erduvan(T.C. MİLLİ EĞİTİM BAKANLIĞI)0000-0001-7254-2296
Pages
914–920
Publication Date
2023-08-31
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The proof based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of this Diophantine equation.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we find that all Mulatu numbers, which are concatenations of two Lucas numbers are 11,17,73,118.
📊
Key Findings & Data:Let 〖(M_k)〗_(k≥0) and 〖(L_k)〗_(k≥0) be the Mulatu and Lucas sequences.
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Conclusion & Impact:The proof based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of this Diophantine equation.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Lucas sayıları · Mulatu sayıları · Logaritmalarda lineer formlar

Abstract

In this paper, we find that all Mulatu numbers, which are concatenations of two Lucas numbers are 11,17,73,118. Let 〖(M_k)〗_(k≥0) and 〖(L_k)〗_(k≥0) be the Mulatu and Lucas sequences. That is, we solve the Diophantine equation M_k=L_m L_n=10^d L_m+L_n in non-negative integers (k,m,n,d), where d denotes the number of digits of L_n. Solutions of this equation are denoted by (k,m,n,d)=(4,1,1,1),(5,1,4,1),(8,4,2,1),(9,1,6,2). In other words, we have the solutions M_4=L_1 L_1=11, M_5=L_1 L_4=17, M_8=L_4 L_2=73, M_9=L_1 L_6=118. The proof based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of this Diophantine equation.

Özet

Bu çalışmada iki Lucas sayısının birleşimi olan tüm Mulatu sayılarının 11,17,73,118 olduğunu buluyoruz. 〖(M_k)〗_(k≥0) ve 〖(L_k)〗_(k≥0) Mulatu ve Lucas dizileri olsun. Yani biz negatif olmayan (k,m,n,d) tam sayılarında M_k=L_m L_n=〖10〗^d L_m+L_n Diyofant denklemini çözüyoruz, burada d, L_n nin basamak sayısını gösterir. Bu denklemin çözümleri (k,m,n,d)=(4,1,1,1),(5,1,4,1),(8,4,2,1),(9,1,6,2) ile ifade edilir. Bir başka deyişle M_4=L_1 L_1=11, M_5=L_1 L_4=17, M_8=L_4 L_2=73, M_9=L_1 L_6=118 çözümlerine sahibiz. İspat Baker’in teorisine dayanmakta ve biz bu denklemi çözmek için logaritmalarda doğrusal formları ve indirgeme metodunu kullandık.

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Cite This Article

Erduvan, F. (2023). Mulatu Numbers That Are Concatenations of Two Lucas Numbers. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, Cilt: 23 Sayı: 4, 914–920. https://doi.org/10.35414/akufemubid.1240679

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