This book contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into five chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2 includes various diophantine equations, some of which are treatable by elementary methods; others are partial solutions of certain unsolved problems. An important method is based on the famous Euler-Bell-Kalm ́ar lemma, with many applications. Article 20 may be considered also as an introduction to Chapter 3 on Arithmetic functions. Here many papers study the famous Smarandache function, the source of inspiration of so many mathematicians or scientists working in other fields. The author has discovered various generalizations, extensions, or analogues functions. Other topics are connected to the composition of arithmetic functions, arithmetic functions at factorials, Dedekind’s or Pillai’s functions, as well as semigroup-valued multiplicative functions. Chapter 4 discusses certain divisibility problems or questions related especially to the sequence of prime numbers.
The author has solved various conjectures by Smarandache, Bencze, Russo etc.; see especially articles 4,5,7,8,9,10. Finally, Chapter 5 studies certain irrationality criteria; some of them giving interesting results on series involving the Smarandache function. Article 3.13 (i.e. article 13 in Chapter 3) is concluded also with a theorem of irrationality on a dual of the pseudo-Smarandache function. A considerable proportion of the notes appearing here have been earlier published in journals in Romania or Hungary (many written in Hungarian or Romanian).
We have corrected and updated these English versions. Some papers appeared already in the Smarandache Notions Journal, or are under publication (see Final References). The book is concluded with an author index focused on articles (and not pages), where the same author may appear more times.
Finally, I wish to express my warmest gratitude to a number of persons and organizations from whom I received valuable advice or support in the preparation of this material. These are the Mathematics Department of the Babe ̧s-Bolyai University, the Domus Hungarica Foundation of Budapest, the Sapientia Foundation of Cluj and also Professors M.L. Perez, B. Crstici, K. Atanassov, P. Haukkanen, F. Luca, L. Panaitopol, R. Sivaramakrishnan, M. Bencze, Gy. Berger, L. T ́oth, V.E.S. Szab ́o, D.M. Miloˇsevi ́c and the late D.S. Mitrinovi ́c. My appreciation is due also to American Research Press of Rehoboth for efficient handling of this publication.