Smarandache Prime Series And Its Related Number Theory Problems

American-Romanian number theorist Florentin Smarandache published a book named "Only problems. Not solutions!" in 1993. In this book, he pre-sented 105 unsolved arithmetical problems and conjectures about special se-quences and functions, many researchers studied these problems.In this dissertation, we used elementary methods and analytical methods to study the Smarandache prime part sequences determinant,the Smarandache Perfect square sequences and the Smarandache double factorial function, se-quences and near pseudo Smarandache function, and given some related nature, asymptotic formulas and solutions of some equations. The main achivements contained in this dissertation are as follows:1. We studied some value properties of the Smarandache prime part se-quences deteminant and discussed two conjectures proposed by professor Wen-peng Zhang:Conjecture 1. For any composite number n≥6, we have the identities c(n)=0 and C(n)=0; Conjecture 2. For any prime q, we have c(q)≠0 and C(q)≠0. About Conjecture 1, we have solved it completely, which will be published in Pure and Applied Mathematics. But for Conjecture 2, it's still an open problem.2. In this section, we defined two functions Sn and In about the Smaran-dache square sequences. We also studied the property of Sn-In as nâ†'∞.3. A special problem of the Smarandache double factorial function of n!is researched. The boundedness of |Sdf((n+1)!) - Sdf(n!)| and Sdf(n!)/n are proved. Moreover, we have founded the solutions of the the Smarandache double factorial function equation.