Friday, February 26, 2016

Review of "A Set of New Smarandache Functions, Sequences and Conjectures in Number Theory," by Felice Russo

Review of

A Set of New Smarandache Functions, Sequences and Conjectures in Number Theory, by Felice Russo ISBN 1879585839

The original Smarandache function was defined as:

If n > 0, S(n) = m is the smallest integer m ≥ 0 such that n divides m!.

 It is a valuable addition to the functions of number theory and much has been written about it.
 Russo opens this book with the definitions of two new functions in number theory that are based on  somewhat similar operations. They are:

The Pseudo-Smarandache totient function Zt(n) is the smallest integer m such that the sum of the values of the Euler totient function φ(k) for k = 1 through m is divisible by n.

The Pseudo-Smarandache-Squarefree function Zw(n) is the smallest integer m such that m^n is divisible by n.

 A series of theorems with proofs describing some of the properties of these functions as well as some problems and conjectures follow the definitions. 

There are many different Smarandache sequences a(n). Many of those sequences are based on the concatenation of numbers. For example, there is the Smarandache repeated digit sequence with 1 end points, where the number of instances of the number is what that number represents. The first few terms of this sequence are:

111, 1221, 13331, 144441, 1555551, 16666661, 177777771, 1888888881, 19999999991, 1101010101010101010101.

 There is a section describing a set of such sequences along with comments about what to study as well as a few problems and conjectures.
The next set of functions are defined for those sequences. For example, the Smarandache Zeta function Sz(s) is the infinite sum from n = 1 through infinity of the ratio 

                    1 / (a(n))^s

where a(n) is an infinite sequence of numbers. Once again, some theorems, problems and conjectures based on these functions are given.
 The next section deals with the Smarandache Double Factorial function Sdf(n)!!. The double factorial function is defined in the following way:

m!! = 1 * 3 * 5 * . . . . * m, if m is odd
m!! = 2 * 4 * 6 * . . . * m, if m is even. 

Sdf(n)!! is defined as the smallest number m such that Sff(n)!! is divisible by m. A large set of theorems with proofs, problems and conjectures regarding this function are then given. The last chapter contains some Smarandache conjectures and unsolved problems. 

Like the best math books that introduce new material, Russo does enough development of the subject so that the reader can see some of the ramifications of the new functions and is motivated to conduct further examination and study. Number theory is one of the oldest branches of mathematics, going all the way back to Diophantus. In this book, some new life is injected into this ancient topic, providing the tinder for further mathematical exploration.

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