6 papers
On Perfect Powers in k-Generalized Pell-Lucas Sequence
Zafer Şiar, Refik Keskin
Let k>=2 and let (Q_{n}^{(k)})_{n>=2-k} be the k-generalized Pell sequence defined by Q_{n}^{(k)}=2Q_{n-1}^{(k)}+Q_{n-2}^{(k)}+...+Q_{n-k}^{(k)} for n>=2 with initial conditions Q_…
A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x^{2}+b^{y}=c^{z}
Refik Keskin, Zafer Şiar
Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine equation x^{2}+b^{y}=c^{z}…
Repdigits in k-generalized Pell sequence
Zafer Şiar, Refik Keskin
Let and let be -generalized Pell sequence defined by \begin{equation*}P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+...+P_{n-k}^{(k)}\end{equati…
On the Exponential Diophantine Equation
Zafer Şiar, Refik Keskin
In this paper, we consider the equation . By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above eq…
A Note On the Exponential Diophantine Equation (a^n-1)(b^n-1)=x^2
Refik Keskin
In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title for all pairs (a,b) such that 1<a<b<101 with some exceptions. There are sixty nine exceptions. In this pa…
On the Diophantine equation F_{n}-F_{m}=2^{a}
Zafer Şiar, Refik Keskin
In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and…