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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