Five years after having published the first volume, the authors obtained many new results related to Smarandaches Problems. They are collected in this book and deal with Smarandache sequences, infinite numbers, functions, formulas, conjunctures in Number Theory.

This book contains 34 papers, most of which were written by participants to the First Northwest Number Theory Conference held in Shangluo Teacher?s College, China, in March, 2005. In this Conference, several professors gave a talk on Smarandache Problems and many participants lectured on them both extensively and intensively. All these papers are original and have been refereed. The themes of these papers range from the mean value or hybrid mean value of Smarandache type functions, the mean value of some famous number theoretic functions acting on the Smarandache sequences, to the convergence property of some infinite series involving the Smarandache type sequences.

The aim of this paper is to study the neutrosophic complex finite rings 𝐶(𝑍𝑛) 𝑎𝑛𝑑 𝐶(< 𝑍𝑛 ∪ 𝐼 >), and to give a classification theorem of these rings. Also, this work introduces full solutions for 12 Kandasamy-Smarandache open problems concerning these structures of generalized rings modulo integers. Also, a necessary and sufficient condition of invertibility in 𝐶(𝑍𝑛) 𝑎𝑛𝑑 𝐶(< 𝑍𝑛 ∪ 𝐼 >) is presented as a partial solution of the famous group of units problem.

A collection of papers concerning Smarandache type functions, numbers, sequences, inteqer algorithms, paradoxes, experimental geometries, algebraic structures, neutrosophic probability, set, and logic, etc.

About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name) and literature, it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. We structured this book using numbered Definitions, Theorems, Conjectures, Notes and Comments, in order to facilitate an easier reading but also to facilitate references to a specific paragraph. We divided the Bibliography in two parts, Writings by Florentin Smarandache (indexed by the name of books and articles) and Writings on Smarandache notions (indexed by the name of authors). We treated, in this book, about 130 Smarandache type sequences, about 50 Smarandache type functions and many solved or open problems of number theory. We also have, at the end of this book, a proposal for a new Smarandache type notion, id est the concept of “a set of Smarandache-Coman divisors of order k of a composite positive integer n with m prime factors”, notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers. This encyclopedia is both for researchers that will have on hand a tool that will help them “navigate” in the universe of Smarandache type notions and for young math enthusiasts: many of them will be attached by this wonderful branch of mathematics, number theory, reading the works of Florentin Smarandache.

Made available online by the Smarandache Notion Journal and the University of New Mexico - Gallup.

A collection of papers concerning Smarandache type functions, numbers, sequences, inteqer algorithms, paradoxes, experimental geometries, algebraic structures, neutrosophic probability, set, and logic, etc.

Papers on an equation involving the Smarandache function and its positive integer solutions, the Smarandache kn-digital subsequence, the Smarandache 3n-digital sequence and the Zhang Wenpeng's conjecture, the quintic supported spline wavelets with numerical integration and similar topics. Contributors: A. A. Majumdar, B. Chen, C. Shi, S. Wang, L. Zhang, A. Saeid, M. Haveshki, T. Veluchamy, P.S.Sivakkumar, and others.