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In this paper, we prove two results related to the solutions of norm form equations. Firstly, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Next, we give a finiteness result concerning solutions of norm form equations representable as sums of $S$-units with a fixed number of terms. To prove these results, we use a deep results concerning the finiteness of the solutions of polynomial-exponential equations and $S$-unit equations., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2403.18924

Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two defined over a function field and $\mathcal{O}_S^*$ be the set of $S$-units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and $S$-units. In particular, we provide a finiteness result for the solutions of the Diophantine equation $U_{n_1} + \cdots + U_{n_r} \in \mathcal{O}_S^*$ in nonnegative integers $n_1, \ldots, n_r$. Furthermore, we study the finiteness result of the Diophantine equation $U_{n_1}^{(1)}+ U_{n_2}^{(2)}+\cdots+U_{n_k}^{(k)} = 0$ in $(n_1, \ldots, n_k)\in \N^k$, where $U_n^{(1)}, \ldots, U_n^{(k)}$ are simple linear recurrence sequences in the function field., Comment: 14 pages

Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $\alpha$ with $\alpha>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n \geq m \geq 0$, $x \geq 2$, and $q \geq 2$. Firstly, we show that there are only finitely many of them for a fixed $x$ using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in $(n, m, x, q)$ with $q, x\geq 2$ under the assumption of the {\em abc-conjecture}. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in $S$-units and Siegel's theorem concerning the finiteness of the number of solutions of a hyperelliptic equation., Comment: 20 pages

Let $S := \{p_1,\ldots ,p_{\ell}\}$ be a finite set of primes and denote by $\mathcal{U}_S$ the set of all rational integers whose prime factors are all in $S$. Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two. In this paper, we provide a finiteness result for the solutions of the Diophantine equation $aU_n + bU_m = z_1 +\cdots +z_r,$ where $n\geq m$ and $z_1, \ldots, z_r\in \mathcal{U}_S$., Comment: Comments are welcome

Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)})_{j\geq 1}$ an increasing sequence composed of integers of the form $U_{n_k} +\cdots + U_{n_1}, \ n_k>\cdots >n_1$. Under certain assumptions, we prove that for any $\epsilon >0,$ there exists an integer $n_{0}$ such that $[U_j^{(k)}]_S < \left(U_j^{(k)}\right)^{\epsilon},$ for $ j > n_0,$ where $[m]_S$ denote the $S$-part of the positive integer $m$. On further assumptions on $(U_{n})_{n \geq 0},$ we also compute an effective bound for $[U_j^{(k)}]_S$ of the form $\left(U_j^{(k)}\right)^{1-c}$, where $c $ is a positive constant depends only on $(U_{n})_{n \geq 0}$ and $S.$

Solid solutions of perovskite oxides (KNbO3)1-x+(La2NiMnO6)x (x=0, 0.1, 0.2 and 0.3) with a variation of band gap (1.33-3.6 eV) have been introduced as a promising photovoltaic absorber. The structural characterization of the prepared samples was carried out using X-ray diffraction (followed by Rietveld refinement) and Raman experiment. As the doping percentage of the monoclinic La2NiMnO6 in the solid-solution increases, the crystal structure of host KNbO3 becomes more symmetric from orthorhombic to cubic. A large reduction in the particle size has also been observed in the solid solutions in comparison to the pure KNbO3. The band gap (~ 1.33 eV) of the synthesized solid solution x=0.1 is found to be very close to the Shockley-Queisser band gap value of 1.34 eV, which suggests the promising photovoltaic possibility in this material. Photoluminescence (PL) emission spectra reveal a strong PL quenching in the solid-solutions in comparison to the KNbO3. The overall structural and optical studies suggest the promising photovoltaic possibility in KNbO3/ La2NiMnO6 solid solution., Comment: 6 pages, 10 Figures, 1 Table

In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + \cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p$ is prime and $x,d,z$ are integers with $1 \leq d \leq 50$.

Let $\{ {U_{n}\}_{n \geq 0} }$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\ldots, p_s\}$ be fixed prime numbers and $\{b_1,\ldots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain the finiteness result for the solution of the Diophantine equation $U_{n_{1}} + \cdots + U_{n_{t}} = b_1 p_1^{z_1} + \cdots+ b_s p_s^{z_s} $ under certain assumptions. Moreover, we explicitly solve the equation $F_{n_1}+ F_{n_2}= 2^{z_1} +3^{z_2}$, in non-negative integers $n_1, n_2, z_1, z_2$ with $z_2\geq z_1$. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker-Davenport reduction method., Comment: 16 pages

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine equation $u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}$ with some conditions on $n_i $ for all $1\leq i \leq t$. Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using the lower bounds for linear forms in logarithms. Further, we use a variant of Baker-Davenport reduction method in Diophantine approximation due to Dujella and Peth\H{o}., Comment: 15 pages

In this paper we have computed all the affine Kac-Moody symmetric spaces which are tame Frechet manifolds starting from the Vogan diagrams related to the affine untwisted Kac-Moody algebras. The detail computation of affine Kac-Moody symmetric spaces associated with A_1^(1) and A_2^(1) are shown algebraically to corroborate our method., Comment: preliminary version

This paper reports on the structural and optical properties of Co-doped TiO2 thin films grown onto (0001) Al2O3 substrates by non-reactive pulsed laser deposition (PLD) using argon as buffer gas. It is shown that by keeping constant the substrate temperature at as low as 310 C and varying only the background gas pressure between 7 Pa and 70 Pa, it is possible to grow either epitaxial rutile or pure anatase thin films, as well as films with a mixture of both polymorphs. The optical band gaps of the films are red shifted in comparison to the values usually reported for undoped TiO2, which is consistent with n-type doping of the TiO2 matrix. Such band gap red shift brings the absorption edge of the Co-doped TiO2 films into the visible region, which might favour their photocatalytic activity. Furthermore, the band gap red shift depends on the films phase composition, increasing with the increase of the Urbach energy for increasing rutile content., Comment: 29 pages, 9 figures, 1 table

Here we report on the structural, optical, electrical and magnetic properties of Co-doped and (Co,Mo)-codoped SnO2 thin films deposited on r-cut sapphire substrates by pulsed laser deposition. Substrate temperature during deposition was kept at 500 C. X-ray diffraction analysis showed that the undoped and doped films are crystalline with predominant orientation along the [101] direction regardless of the doping concentration and doping element. Optical studies revealed that the presence of Mo reverts the blue shift trend observed for the Co-doped films. For the Co and Mo doping concentrations studied, the incorporation of Mo did not contribute to increase the conductivity of the films or to enhance the ferromagnetic order of the Co-doped films., Comment: 18 pages, 5 figures, 1 table