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Three Diophantine equations concerning the polygonal numbers [PDF]
Notes on Number Theory and Discrete MathematicsMany authors investigated the problem about the linear combination of two polygonal numbers being a perfect square, i.e., the Diophantine equation mPₖ(x)+nPₖ(y)=z², where Pₖ(x) denotes the x-th k-polygonal number and m, n are positive integers.
Yong Zhang, Mei Jiang, Qiongzhi Tang
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Some experiments with Ramanujan-Nagell type Diophantine equations [PDF]
, 2014 Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many ...
Ulas, Maciej
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Separable Diophantine Equations [PDF]
Transactions of the American Mathematical Society, 1945 Theoretically, as noted by Skolem [3, p. 21(1), the general problem of algebraic diophantine analysis is reducible to the case in which occur only equations and inequalities of degree not higher than the second. For the extensive class of separable systems defined in ?6, this reduction can be performed effectively, eventuating in the complete integer ...
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On the size of Diophantine m-tuples in imaginary quadratic number rings
Bulletin of Mathematical Sciences, 2021 A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers.
Nikola Adžaga
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Diophantine equations involving factorials [PDF]
Mathematica Bohemica, 2017 We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only
Horst Alzer, Florian Luca
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Arrested development and fragmentation in strongly-interacting Floquet systems
SciPost Physics, 2023 We explore how interactions can facilitate classical like dynamics in models with sequentially activated hopping. Specifically, we add local and short range interaction terms to the Hamiltonian and ask for conditions ensuring the evolution acts as a ...
Matthew Wampler, Israel Klich
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Multiplicative Diophantine equations
Journal of Number Theory, 1992 The solution of the diophantine equation \(\prod_{i=1}^ n x_ i= \prod_{i=1}^ n y_ i\) is given in terms of \(n^ 2\) parameters (Bell's theorem) [cf. the first author, Proc. Ramanujan Cent. Int. Conf., Annamalainagar/India 1987, RMS Publ. 1, 141-146 (1988; Zbl 0696.10014)].
Srinivasa Rao, K. +2 more
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Sparse solutions of linear Diophantine equations [PDF]
, 2016 We present structural results on solutions to the Diophantine system $A{\boldsymbol y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number of non-zero entries.
Aliev, Iskander +3 more
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Quiver theories, soliton spectra and Picard-Lefschetz transformations [PDF]
, 2002 Quiver theories arising on D3-branes at orbifold and del Pezzo singularities are studied using mirror symmetry. We show that the quivers for the orbifold theories are given by the soliton spectrum of massive 2d N=2 theory with weighted projective spaces ...
A. Hanany +44 more
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Glasgow Mathematical Journal, 1985 I was recently challenged to find all the cases when the sum of three consecutive integral cubes is a square; that is to find all integral solutions x, y ofy2=(x−1)3+x3+(x+1)3=3x(x2+2)This is an example of a curve of genus 1. There is an effective procedure for finding all integral points on a given curve of genus 1 ([1, Theorem 4.2], [2]): that is, it
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