• REFIK KESKIN

Articles written in Proceedings – Mathematical Sciences

• Positive Integer Solutions of the Diophantine Equation $x^2 - L_n xy + (-1)^n y^2 = \pm 5^r$

In this paper, we consider the equation $x^2-L_n xy+(-1)^n y^2=\pm 5^r$ and determine the values of 𝑛 for which the equation has positive integer solutions 𝑥 and 𝑦. Moreover, we give all positive integer solutions of the equation $x^2-L_n xy+(-1)^n y^2=\pm 5^r$ when the equation has positive integer solutions.

• A note on the exponential diophantine equation $(a^{n} − 1)(b^{n} − 1) = x^{2}$

In 2002, Luca and Walsh (J. Number Theory 96 (2002) 152–173) solved the diophantine equation for all pairs $(a, b)$ such that $2\leq a$ < $b\leq 100$ with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation $(a^{n}−1)(b^{n}−1) = x^{2}$. It is also proved that this equation has no solutions if $a, b$ have opposite parity and $n$ >$4$ with $2|n$. Here, the equation is also solved for the pairs $(a, b) = (2, 50), (4, 49), (12, 45), (13, 76), (20, 77), (28, 49), (45, 100)$. Lastly, we show that when b is even, the equation $(a^{n} − 1)(b^{2n}a^{n} − 1) = x^{2}$ has no solutions $n, x$.

• On solutions of the diophantine equation $F_{n} − F_{m} = 3^{a}$

In this paper, we find non-negative ($n, m, a$) integer solutions of thediophantine equation $F_{n} − F_{m} = 3^{a}$, where $Fn$ and $Fm$ are Fibonacci numbers. For proving our theorem, we use lower bounds in linear forms.

• Repdigits as products of two Fibonacci or Lucas numbers

In this study, we show that if $2 \leq m \leq n$ and $F_{m} F_{n}$ represents a repdigit, then $(m, n)$ belongs to the set $$\{(2, 2), (2, 3), (3, 3), (2, 4), (3, 4), (4, 4), (2, 5), (2, 6), (2, 10)\}.$$ Also, we show that if $0 \leq m \leq n$ and $L_{m} L_{n}$ represents a repdigit, then $(m, n)$ belongs to the set $$\left\{ \begin{array}{l}(0, 0), (0, 1), (1, 1), (0, 2), (1, 2), (2, 2), (0, 3),\\ (1, 3), (1, 4), (1, 5), (2, 5), (3, 5), (4, 5) \end{array}\right\}$$

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019