Volume 55, Issue 1
Repdigits Base $b$ as Difference of Two Fibonacci Numbers

J. Math. Study, 55 (2022), pp. 84-94.

Published online: 2022-01

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• Abstract

In this paper, we find all repdigits expressible as difference of two Fibonacci numbers in base $b$ for $2\leq b\leq10.$ The largest repdigits in base $b$, which can be written as difference of two Fibonacci numbers are \begin{align*}&F_{9}-F_{4}=34-3=31=(11111)_{2},~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(111111)_{3},\\&F_{14}-F_{7}=377-13=364=(222)_{4},~~ \text{ }F_{9}-F_{4}=34-3=31=(111)_{5},\\&F_{11}-F_{4}=89-3=86=(222)_{6},~~~~~~~~\text{ }F_{13}-F_{5}=233-5=228=(444)_{7},\\&F_{10}-F_{2}=55-1=54=(66)_{8},~~~~~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(444)_{9},\end{align*} and $$F_{15}-F_{10}=610-55=555=(555)_{10}.$$

As a result, it is shown that the largest Fibonacci number which can be written as a sum of a repdigit and a Fibonacci number is $F_{15}=610=555+55=555+F_{10}.$

• Keywords

Fibonacci numbers, repdigit, Diophantine equations, linear forms in logarithms.

11B39, 11J86, 11D61

zsiar@bingol.edu.tr (Zafer Şiar)

erduvanmat@hotmail.com.tr (Fatih Erduvan)

keskin@sakarya.edu.tr (Refik Keskin)

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@Article{JMS-55-84, author = {Zafer and Şiar and zsiar@bingol.edu.tr and 22009 and Bingöl University, Department of Mathematics, Bingöl 12000, Turkey and Zafer Şiar and Fatih and Erduvan and erduvanmat@hotmail.com.tr and 22010 and Sakarya University, Department of Mathematics, Sakarya TR-54187, Turkey and Fatih Erduvan and Refik and Keskin and keskin@sakarya.edu.tr and 22011 and Sakarya University, Department of Mathematics, Sakarya TR-54187, Turkey and Refik Keskin}, title = {Repdigits Base $b$ as Difference of Two Fibonacci Numbers}, journal = {Journal of Mathematical Study}, year = {2022}, volume = {55}, number = {1}, pages = {84--94}, abstract = {

In this paper, we find all repdigits expressible as difference of two Fibonacci numbers in base $b$ for $2\leq b\leq10.$ The largest repdigits in base $b$, which can be written as difference of two Fibonacci numbers are \begin{align*}&F_{9}-F_{4}=34-3=31=(11111)_{2},~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(111111)_{3},\\&F_{14}-F_{7}=377-13=364=(222)_{4},~~ \text{ }F_{9}-F_{4}=34-3=31=(111)_{5},\\&F_{11}-F_{4}=89-3=86=(222)_{6},~~~~~~~~\text{ }F_{13}-F_{5}=233-5=228=(444)_{7},\\&F_{10}-F_{2}=55-1=54=(66)_{8},~~~~~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(444)_{9},\end{align*} and $$F_{15}-F_{10}=610-55=555=(555)_{10}.$$

As a result, it is shown that the largest Fibonacci number which can be written as a sum of a repdigit and a Fibonacci number is $F_{15}=610=555+55=555+F_{10}.$

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v55n1.22.07}, url = {http://global-sci.org/intro/article_detail/jms/20196.html} }
TY - JOUR T1 - Repdigits Base $b$ as Difference of Two Fibonacci Numbers AU - Şiar , Zafer AU - Erduvan , Fatih AU - Keskin , Refik JO - Journal of Mathematical Study VL - 1 SP - 84 EP - 94 PY - 2022 DA - 2022/01 SN - 55 DO - http://doi.org/10.4208/jms.v55n1.22.07 UR - https://global-sci.org/intro/article_detail/jms/20196.html KW - Fibonacci numbers, repdigit, Diophantine equations, linear forms in logarithms. AB -

In this paper, we find all repdigits expressible as difference of two Fibonacci numbers in base $b$ for $2\leq b\leq10.$ The largest repdigits in base $b$, which can be written as difference of two Fibonacci numbers are \begin{align*}&F_{9}-F_{4}=34-3=31=(11111)_{2},~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(111111)_{3},\\&F_{14}-F_{7}=377-13=364=(222)_{4},~~ \text{ }F_{9}-F_{4}=34-3=31=(111)_{5},\\&F_{11}-F_{4}=89-3=86=(222)_{6},~~~~~~~~\text{ }F_{13}-F_{5}=233-5=228=(444)_{7},\\&F_{10}-F_{2}=55-1=54=(66)_{8},~~~~~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(444)_{9},\end{align*} and $$F_{15}-F_{10}=610-55=555=(555)_{10}.$$

As a result, it is shown that the largest Fibonacci number which can be written as a sum of a repdigit and a Fibonacci number is $F_{15}=610=555+55=555+F_{10}.$

Zafer Şiar, Fatih Erduvan & Refik Keskin. (2022). Repdigits Base $b$ as Difference of Two Fibonacci Numbers. Journal of Mathematical Study. 55 (1). 84-94. doi:10.4208/jms.v55n1.22.07
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