Quadratic Diophantine Equations

Andreescu, Titu; Andrica, Dorin

doi:10.1007/978-0-387-54109-9

Cited by 32 publications

(30 citation statements)

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“…Further information and additional proofs may be obtained from the mathematical literature [204], [205], [206].…”

Section: E Details Of Number Theorymentioning

confidence: 99%

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Crystallography and Surface Structure: An Introduction for Surface Scientists and Nanoscientists

Hermann

2016

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Book cover text: "The book cover includes a scanning electron miroscopy picture of a stepped Cu 2 O surface (large circular area) kindly provided by Marc Willinger, Fritz-Haber Institute Berlin." 2 PREFACE TO THE SECOND EDITIONAs a result of feedback from readers of the first edition of this book and from colleagues, this second edition is a major revision which goes beyond mere error correction and minor clarification.While the book has been modified and extended greatly, the initial concept of a combined tutorial introduction into surface crystallography, with bulk crystallography as a basis, and an overview of modern subjects for the advanced researcher is still conserved. Many sections have been updated for completeness and extended to include recent developments due to the advent of new and more refined measuring techniques.The second edition is also targeted at researchers working on graphene and other weakly adsorbing overlayers which form large size moiré patterns observed by scanning tunneling and electron microscopy. They might appreciate the new section on moiré lattice formation which until now has not been available in textbook format. Nanoparticle physicists and materials scientists who are interested in structure information of very small particles and seek to connect e.g. electronic and magnetic properties with structural data, may benefit from the sections about nanoparticles, crystal spheres, nanotubes, as well as faceting. This might also interest catalytic chemists trying to interpret chemical behavior such as reactivity by structural information of small particles.Specific items which have been newly added or revised include -nanoclusters and crystallites, giving a basic overview on structure details, -incommensurate and quasicrystals, being treated on a common basis, -basics of epitaxy and crystal growth, -further details on chiral surfaces and adsorbates, -the theoretical treatment of high-order commensurate (HOC) overlayers, -the theory of interference lattices and moiré patterns, -the geometric structure of high-symmetry adsorbate sites, -more detailed computational algorithms in the appendices, -structure database formats, documenting measured surface structures.Furthermore, the list of references to original publications and books on specific subjects has been revised and extended to account for more recent experimental and theoretical developments. The set of exercises concluding each section has been substantially enlarged following suggestions by readers of the first book edition. All structure graphics in this book have been created using the interac- PREFACE TO THE FIRST EDITIONThe objective of this book is to provide students and researchers with the crystallographic foundations necessary to understand structure and symmetry of surfaces and interfaces of crystalline materials. This includes macroscopic single crystals as well as crystalline nanoparticles. Knowledge of their geometric properties is a prerequisite for the interpretation of corresponding experimental and theoretica...

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“…Further information and additional proofs may be obtained from the mathematical literature [204], [205], [206].…”

Section: E Details Of Number Theorymentioning

confidence: 99%

“…There is an extensive literature on quadratic Diophantine equations [204], [205] which will not be reviewed in this appendix. Here we focus only on solutions of specific equations which arise in connection with neighbor shells in crystals with high symmetry and overlayer reconstruction at single crystal surfaces.…”

Section: E4 Quadratic Diophantine Equationsmentioning

confidence: 99%

Crystallography and Surface Structure: An Introduction for Surface Scientists and Nanoscientists

Hermann

2016

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“…Further the authors conjectured that only the quadruple (k, r, p, q) = (4, 3, 8, 2) of positive integers satisfies (2). The conjecture was completely justified by Alvarado et al [1].…”

Section: Introductionmentioning

confidence: 94%

“…, and solved it for the cases (p, q) = (2, 1), (3,1), (3,2), and for 2 ≤ p ≤ q by showing the non-existence of any solution. Further the authors conjectured that only the quadruple (k, r, p, q) = (4, 3, 8, 2) of positive integers satisfies (2).…”

Section: Introductionmentioning

confidence: 99%