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Chapter I INTRODUCTION Chapter I INTRODUCTION INTRODUCTION Number Theory, usually referred to as the Queen of Mathematics is one of the oldest, the largest and the fascinating branches of Pure Mathematics. It has been a matter of attraction to various mathematicians and to the lovers of Mathematics since antiquity. Probably, the mankind has been fascinated by numbers (integers) since discovery of counting. In other words, human beings have been able to get the excitement and thrill of Numbers. Throughout history, numbers have a tremendous influence on our culture and on our language[6,8,16,65,70].“When I considered what people generally want in calculating, I found that it always is a number”, said Mohammed Ben Musa Al-Khawarizmi. (From the treasure of Mathematics, P.420; H.O.Midonich, Philosophical Library, 1965). Number is the essence of mathematical calculation. The ancient Greeks established it as a subject which they called Arithmetic. Until the middle of the twentieth century, Number Theory was considered to be the purest area of Mathematics and least likely sullied by applications. Now the subject is an exciting mix of ideas pursued for their own interest and a rather exotic variety of applications [2,9,78,80,83,85,91,92,93,94,95]. Every layman can appreciate the excitement of Mathematics by playing systematically with integers and geometrical forms, discovering patterns in them and trying to prove more. In fact, Number Theory is the great and rich intellectual heritage of mankind and is essentially a man made world to meet his ideals of intellectual perfection wherein we discover beautiful patterns in numbers and space. Integers exhibit fascinating properties, they form sequences and they form patterns. Recognizing number patterns is also an important problem solving skill. Working with number patterns leads to the concept of functions in Mathematics, a formal description of the 1 Chapter I INTRODUCTION relationships among different quantities. The greatest fascination lies in the limitless supply of exciting, non-routine and challenging problems they have provided [1,3,4,5,7,10,11,12,13,14,25,28,29,31,33,35,67,68,69,74,75,76,77,84,87, 88,89,90]. Geometric integer sequences are the significant integers that can be represented as geometrical forms such as triangle, square, pentagon, hexagon, centered hexagon, star and so on prominently found in scriptures. On-Line Encyclopedia of Integer Sequences is an excellent site that has a huge searchable database of such integer sequences. The Mathematics of the Pythagorean times covers many of the topics in algebra as well as in geometry. The foundations for number theory, as a discipline were laid by the Greek mathematician Pythagoras and his disciples known as the Pythagoreans. The Pythagorean brotherhood believed that “everything is number” and some numbers have mystical powers. The Pythagoreans have been credited with the invention of amicable numbers, perfect numbers, figurate numbers and Pythagorean triples or Pythagorean Triangles. Since there is a one-one correspondence between Pythagorean triples and Pythagorean Triangles, we may use them interchangeably. Particularly, the study of Pythagorean triples is fascinating. They have been studied from different angles and many methods of generating Pythagorean triples have been proposed in [17,20,21,22,23,26,27,36,58,59,60,61,62,63,82,86,96]. In fact, Number Theory is a treasure house in which the search for many hidden relations and properties among numbers form a treasure hunt. A bit of attention at various times on the branch of Pythagorean mathematics and their relationships with many areas of mathematics have been carried due to the fact that they are figurate numbers which according to the Pythagorean brotherhood was a geometry of aggregates of points but not of lines as in Euclidean geometry. The numbers that can be represented by a regular geometric arrangement of equally spaced points are called 2 Chapter I INTRODUCTION figurate numbers or polygonal numbers. A few examples of polygonal numbers are Triangular, Pentagonal, Hexagonal, Octagonal, Heptagonal, Decagonal and so on.Some basic concepts and properties among a few polygonal numbers are presented in [15,18,19,24,30,32,34,64,66,71,72,73,79,81]. This has been a topic of keen interest to many Mathematicians worldwide for several centuries because of its historical importance and they offer good applications in the fields of Pattern Classification, Graph Theory and so on. In particular centered Hexagonal numbers have practical applications in Materials Logistics Management such as, in packing round items into larger round containers and Vienna Sausages into round cans. The study of figurate numbers continues to be a source of interest to both amateur and professional Mathematicians as they can be studied both algebraically and geometrically. These numbers have motivated us to search for new varieties of patterns of special numbers. It is therefore towards this end, we search for interesting properties among different patterns in two dimensional figurate (polynomial) numbers, three dimensional figurate (pyramidal) numbers,four dimensional figurate numbers and special numbers such as Kynea numbers, Schröder numbers, Magna numbers and so on. This dissertation consists of 7 chapters. Chapter I provides the historical background and necessary literature survey for the varieties of number patterns. In Chapters II-VII, different number patterns and their related properties are presented. In Chapter II, special patterns of Pythagorean triangles are analyzed in Sections (A), (B) and (C). In Section (A) [37], the patterns of Pythagorean triangles for each of which one leg exceeds k times the other leg by an arbitrary positive constant are analyzed. Also, patterns of Pythagorean triangles, where, in 3 Chapter I INTRODUCTION each of which, the hypotenuse is k times a leg added with an arbitrary non-zero integer, are presented in appendix. Section (B) [44] generates patterns of pairs of Pythagorean triangles with equal perimeters. A few examples are also illustrated. As the polygonal numbers form a branch of Pythagorean Mathematics, in Section (C) [45] varieties of relation between the legs of Pythagorean triangle and special polygonal numbers are exhibited in a tabular form. Chapter III consists of three Sections (A), (B) and (C). Section (A) [41] deals star numbers whose base is a hexagon. The recurrence relation among the star square numbers is presented. The ranks of star numbers, such that each star number decreased by unity representing a square integer or twelve times a square integer are obtained. Section (B) [39] concerns with some interesting observations on centered hexagonal numbers. In particular, the ranks of three centered hexagonal numbers in arithmetic progression are obtained. Explicit formulas for the ranks of centered hexagonal numbers which are simultaneously equal to Nanogonal numbers are presented. In Section(C) [43], formulas for obtaining the ranks of centered hexagonal numbers which are simultaneously equal to centered triangular, centered square, centered pentagonal, centered heptagonal, centered octagonal, centered nanogonal, centered decagonal, centered hendecagonal, centered dodecagonal, centered tridecagonal, centered tetradecagonal, centered pentadecagonal,centered hexadecagonal, centered heptadecagonal, centered octadecagonal, centered nanodecagonal, centered icosagonal numbers are presented. The corresponding recurrence relations satisfied by the ranks are also given. Chapter IV consists of 4 Sections (A), (B), (C) and (D). In Section (A) [40], generalized integer sequence on Jacobsthal numbers is generated from the 4 Chapter I INTRODUCTION =β + α α≠ β αβ > recurrence relation Jn+2 J n + 1 J n , ( , 0 ) with the initial = = conditions J0 aJ , 1 b where a, b not zeros simultaneously. Various relations among the members of the above sequence are given. It can be noted that, for the particular values of a and b , the sequences of Jacobsthal numbers and Jacobsthal Lucas numbers are deduced. In Section (B) [38], different sequences of numbers are generated from the =++α β γ αβγ > recurrence relation Pn+3 P n + 2 P n + 1 P n ( ,, 0 ) with initial = = = conditions P0 AP , 1 BP , 2 C assuming different values for A , B and C . Particularly, Perrin, Padovan and Sofo integer sequences are deduced and in each case, a few interesting relations are given. Also, three more new integer sequences are deduced. In Section (C) [46], an explicit formula representing the general term of the sequence whose members are generated from the difference equation TRIn( +=39) TRIn( +− 226) TRIn( ++ 124) TRIn( ) with initial conditions TRI (1) = 1 ,TRI(2) = 2 , TRI (3) = 3 . A few interesting relations among the members of the above sequence are given. In Section (D) [47], explicit formulae representing the general terms for the following two sequences given by: (i) A( n ) : 2,9,50,289,1682,... 16 196 2704 (ii) B( n ):1, ,9, ,121, ,1681,... 6 6 6 are obtained. Various relations among the elements of the sequences {A( n ) } and {B( n ) } are correspondingly exhibited. Chapter V consisting of 4 Sections (A), (B), (C) and (D) deals with three dimensional figurate numbers. The first three sections are concerned with 5 Chapter I INTRODUCTION Pyramidal numbers and the fourth section is on Stella Octangular Numbers whose base polygon is an octahedral number. Section (A) [48] concerns with triangular pyramidal (tetrahedral) number. Varieties of interesting relations among these numbers are exhibited. In Section (B) [52] relations between Pentagonal Pyramidal numbers of successive ranks have been analyzed. Also, some identities relating pentagonal pyramidal numbers and Jacobsthal numbers are presented. In Section (C) [53], a few interesting relations between the hexagonal pyramidal numbers and other special pyramidal numbers are presented. In Section (D) [57], a fascinating three dimensional figurate number called Stella Octangular Number which is constructed by adjoining tetrahedral numbers to all the eight faces of an octahedral number is analyzed. Various interesting relations between Stella Octangular numbers and figurate numbers of dimensions two, three and four are illustrated.
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