Mathematics

The  following  problem is published in the November 2014 issue of the problem solving journal CRUX  MATHEMATICORUM published  by  the Canadian Mathematical Society; specifically in  issue No 9 , Volume 40; on page 368. The problem at hand is
Contest  Corner Problem CC142,
Problem1(ProblemCC142) : Roboto  writes down a number.
Every minute , he increases the existing number by double of
the number of its natural divisors(including1 and itself).
For example , if he started with 5 , the sequence would be
5 , 9 , 15 , 23 , ...  What is the maximum number of perfect
squares  that appear on the board within 24  hours?
This  paper  is 9 handwritten pages long , We present a solution
to this  problem in Section2 , pages 3 through 9.
We  show that the above maximum number is either zero or one.  Note that such a sequence( as described in Problem1)
contains  (24)(60)  terms; since each hour contains 60 minutes.
Result6 of this paper, on page9 , offers a comprehensive solution to this  problem :
Result6 : Let  a(1)  , a(2) , .... , a( k) ,  be a finite sequence (with at least
two terms)  be a finite  sequence of positive integers defined
recursively  as  follows:
  a(1)= a(1) ;  and  a(I+1) = a( i )  +  2f(a(i))  ; for I = 1 , ... , k-1 .
Where  f( a(I) )  denotes the number of positive integer divisors
of  a(I)  ;  including  1  and  a(I) .
Then:
(i)  If  the  first term a(1)  is  congruent  to 2 or 3 modulo4 ; none
of  the k  terms of the above sequence is  a  perfect square.
(I)  If  the first  term a(1) is congruent to 0 or 1(mod4) .
Then  the  above  sequence contains  at  most one term which is  a perfect  square.
For  example ,consider  the  four  sequences  each containing
four  terms; and with  the first term being congruent to 0 or 1
modulo4:
13 , 17 , 21 , 29;
5 , 9 , 15 , 23 ;
8 , 16 , 26 , 34 ;
12 , 24 , 40 , 56 .

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The Abstract
The introductory course aims to facilitate understanding of subtle points in description, properties and applications of special functions which are commonly met in the modern problems of applied mathematics and physics. In addition, the course provides overview of major mathematical textbooks, databases and supplementary materials in order to find a quick and efficient way of solving a particular problem involving special functions. The course book is based on the lecture series which being given to MSc and PhD students in applied mathematics and physics. The main features of the book are simplified description and an effective usage of several main concepts and previous operations tend to special functions. The particular attention has been drawn to the practical applications of special functions in analytical models of modern physics. The ultimate goal of the authors is to illustrate the importance of independent research work alongside with creativity in developing of an effective analytical method to facilitate understanding of the particular physics phenomenon.
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