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On Perfect Powers in k-Generalized Pell-Lucas Sequence

Autor: Şiar, Zafer, Keskin, Refik

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_{-(k-2)}^{(k)}=Q_{-(k-3)}^{(k)}=...=Q_{-1}^{(k)}=0, Q_{0}^{(k)}=2,Q_{1

Externí odkaz: http://arxiv.org/abs/2209.04190

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A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x^{2}+b^{y}=c^{z}

Autor: Keskin, Refik, Şiar, Zafer

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} has only the positive integer solutions (x,y,z)=(a,2,2). In this study

Externí odkaz: http://arxiv.org/abs/2105.14814

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Repdigits in k-generalized Pell sequence

Autor: Şiar, Zafer, Keskin, Refik

Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be $k$-generalized Pell sequence defined by \begin{equation*}P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+...+P_{n-k}^{(k)}\end{equation*} for $n\geq 2$ with initial conditions \begin{equation*}P_{-(k-2)}

Externí odkaz: http://arxiv.org/abs/2009.13387

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An exponential Diophantine equation related to the difference of powers of two Fibonacci numbers

Autor: Şiar, Zafer

In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number.

Externí odkaz: http://arxiv.org/abs/2002.03783

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On the Exponential Diophantine Equation $(a^2-2)(b^2-2)=x^2$

Autor: Şiar, Zafer, Keskin, Refik

In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are different

Externí odkaz: http://arxiv.org/abs/1801.04770

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On the Diophantine equation F_{n}-F_{m}=2^{a}

Autor: Şiar, Zafer, Keskin, Refik

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 and a version of the Baker-Davenport reduction method in diophantine a

Externí odkaz: http://arxiv.org/abs/1712.10138

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