Araştırma Makalesi
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İki Lucas Sayısının Birleşimi Olan Mulatu Sayıları

Yıl 2023, Cilt: 23 Sayı: 4, 914 - 920, 31.08.2023
https://doi.org/10.35414/akufemubid.1240679

Öz

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.

Kaynakça

  • Alan, M., 2022. On Concatenations of Fibonacci and Lucas Numbers. Bulletin of the Iranian Mathematical Society, 48(5), 2725-2741.
  • Altassan, A., and Alan, M., 2022. On Mixed Concatenations of Fibonacci and Lucas Numbers Which are Fibonacci Numbers. arxiv preprint arxiv:2206.13625, arxiv.org.
  • Annouk, I., and Özer, Ö., 2022. New significant results on Fermat numbers via elementary arithmetic methods. Theoretical Mathematics & Applications, 12(3), 1-10.
  • Badidja, S., Mokhtar, A.A., and Özer, Ö., 2021. Representation of Integers by k- Generalized Fibonacci Sequences and Applications in Cryptography. Asian-European Journal of Mathematics, 14(9), 2150157.
  • Banks, W.D., and Luca, F., 2005. Concatenations with binary reccurent sequences. Journal of Integer Sequences, 8(5), 1-3.
  • Bugeaud, Y., Mignotte, M., and Siksek, S., 2006. Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematics, 163(3), 969-1018.
  • Bugeaud, Y., 2018. Linear Forms in Logarithms and Applications. IRMA Lectures in Mathematics and Theoretical Physics 28, Zurich, European Mathematical Society, 1-176.
  • Bravo, J.J., Gomez, C.A., and Luca, F., 2016. Powers of two as sums of two k-Fibonacci numbers. Miskolc Mathematical Notes, 17(1), 85-100.
  • De Weger, B.M.M., 1989. Algorithms for Diophantine Equations. CWI Tracts 65. Stichting Mathematisch Centrum, Amsterdam, 1-69.
  • Deza, E., 2021. Mersenne Numbers and Fermat Numbers, ISSBN: 9811230315, 9789811230318, WSPC. 1-328.
  • Erduvan, F., 2023. Lucas numbers which are are concatenations of two Lucas numbers. Hodja Akhmet Yassawi 7. International Congress on Scientific Research Mingachevir State University, Mingachevir, Azerbaijan, February 24-25, 3, 176-181.
  • Kılıç, E., Taşçı, D., 2006. On the generalized order-k Fibonacci and Lucas numbers. Journal of Mathematics, 36(6), 1915-1926.
  • Lemma, M., 2011. The Mulatu Numbers. Advances and Applications in Mathematical Sciences, 10(4), 431-440.
  • Schmidt, W.M., 1991. Diophantine Approximations and Diophantine Equations. Springer. 34-72.
  • Tichy, R.F., Schlickewei, H.P., and Schmidt, K., 2008. Diophantine Approximation: Festschrift for Wolfgang Schmidt Developments in Mathematics. Springer. 1-413.
  • Zannier, U., 2003. Some Applications of Diophantine Approximation to Diophantine Equations: With Special Emphasis on the Schmidt Subspace Theorem. Forum. 1-69.

Mulatu Numbers That Are Concatenations of Two Lucas Numbers

Yıl 2023, Cilt: 23 Sayı: 4, 914 - 920, 31.08.2023
https://doi.org/10.35414/akufemubid.1240679

Öz

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.

Kaynakça

  • Alan, M., 2022. On Concatenations of Fibonacci and Lucas Numbers. Bulletin of the Iranian Mathematical Society, 48(5), 2725-2741.
  • Altassan, A., and Alan, M., 2022. On Mixed Concatenations of Fibonacci and Lucas Numbers Which are Fibonacci Numbers. arxiv preprint arxiv:2206.13625, arxiv.org.
  • Annouk, I., and Özer, Ö., 2022. New significant results on Fermat numbers via elementary arithmetic methods. Theoretical Mathematics & Applications, 12(3), 1-10.
  • Badidja, S., Mokhtar, A.A., and Özer, Ö., 2021. Representation of Integers by k- Generalized Fibonacci Sequences and Applications in Cryptography. Asian-European Journal of Mathematics, 14(9), 2150157.
  • Banks, W.D., and Luca, F., 2005. Concatenations with binary reccurent sequences. Journal of Integer Sequences, 8(5), 1-3.
  • Bugeaud, Y., Mignotte, M., and Siksek, S., 2006. Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematics, 163(3), 969-1018.
  • Bugeaud, Y., 2018. Linear Forms in Logarithms and Applications. IRMA Lectures in Mathematics and Theoretical Physics 28, Zurich, European Mathematical Society, 1-176.
  • Bravo, J.J., Gomez, C.A., and Luca, F., 2016. Powers of two as sums of two k-Fibonacci numbers. Miskolc Mathematical Notes, 17(1), 85-100.
  • De Weger, B.M.M., 1989. Algorithms for Diophantine Equations. CWI Tracts 65. Stichting Mathematisch Centrum, Amsterdam, 1-69.
  • Deza, E., 2021. Mersenne Numbers and Fermat Numbers, ISSBN: 9811230315, 9789811230318, WSPC. 1-328.
  • Erduvan, F., 2023. Lucas numbers which are are concatenations of two Lucas numbers. Hodja Akhmet Yassawi 7. International Congress on Scientific Research Mingachevir State University, Mingachevir, Azerbaijan, February 24-25, 3, 176-181.
  • Kılıç, E., Taşçı, D., 2006. On the generalized order-k Fibonacci and Lucas numbers. Journal of Mathematics, 36(6), 1915-1926.
  • Lemma, M., 2011. The Mulatu Numbers. Advances and Applications in Mathematical Sciences, 10(4), 431-440.
  • Schmidt, W.M., 1991. Diophantine Approximations and Diophantine Equations. Springer. 34-72.
  • Tichy, R.F., Schlickewei, H.P., and Schmidt, K., 2008. Diophantine Approximation: Festschrift for Wolfgang Schmidt Developments in Mathematics. Springer. 1-413.
  • Zannier, U., 2003. Some Applications of Diophantine Approximation to Diophantine Equations: With Special Emphasis on the Schmidt Subspace Theorem. Forum. 1-69.
Toplam 16 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Fatih Erduvan 0000-0001-7254-2296

Erken Görünüm Tarihi 29 Ağustos 2023
Yayımlanma Tarihi 31 Ağustos 2023
Gönderilme Tarihi 22 Ocak 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 23 Sayı: 4

Kaynak Göster

APA Erduvan, F. (2023). Mulatu Numbers That Are Concatenations of Two Lucas Numbers. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 23(4), 914-920. https://doi.org/10.35414/akufemubid.1240679
AMA Erduvan F. Mulatu Numbers That Are Concatenations of Two Lucas Numbers. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. Ağustos 2023;23(4):914-920. doi:10.35414/akufemubid.1240679
Chicago Erduvan, Fatih. “Mulatu Numbers That Are Concatenations of Two Lucas Numbers”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 23, sy. 4 (Ağustos 2023): 914-20. https://doi.org/10.35414/akufemubid.1240679.
EndNote Erduvan F (01 Ağustos 2023) Mulatu Numbers That Are Concatenations of Two Lucas Numbers. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 23 4 914–920.
IEEE F. Erduvan, “Mulatu Numbers That Are Concatenations of Two Lucas Numbers”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, c. 23, sy. 4, ss. 914–920, 2023, doi: 10.35414/akufemubid.1240679.
ISNAD Erduvan, Fatih. “Mulatu Numbers That Are Concatenations of Two Lucas Numbers”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 23/4 (Ağustos 2023), 914-920. https://doi.org/10.35414/akufemubid.1240679.
JAMA Erduvan F. Mulatu Numbers That Are Concatenations of Two Lucas Numbers. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2023;23:914–920.
MLA Erduvan, Fatih. “Mulatu Numbers That Are Concatenations of Two Lucas Numbers”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, c. 23, sy. 4, 2023, ss. 914-20, doi:10.35414/akufemubid.1240679.
Vancouver Erduvan F. Mulatu Numbers That Are Concatenations of Two Lucas Numbers. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2023;23(4):914-20.