Research Article
BibTex RIS Cite

İki Lucas Sayısının Birleşimi Olan Mulatu Sayıları

Year 2023, , 914 - 920, 31.08.2023
https://doi.org/10.35414/akufemubid.1240679

Abstract

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.

References

  • 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

Year 2023, , 914 - 920, 31.08.2023
https://doi.org/10.35414/akufemubid.1240679

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.

References

  • 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.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Fatih Erduvan 0000-0001-7254-2296

Early Pub Date August 29, 2023
Publication Date August 31, 2023
Submission Date January 22, 2023
Published in Issue Year 2023

Cite

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. August 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, no. 4 (August 2023): 914-20. https://doi.org/10.35414/akufemubid.1240679.
EndNote Erduvan F (August 1, 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, vol. 23, no. 4, pp. 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 (August 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, vol. 23, no. 4, 2023, pp. 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.


Bu eser Creative Commons Atıf-GayriTicari 4.0 Uluslararası Lisansı ile lisanslanmıştır.