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Recurrence Relation in Number Theory International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893 RECURRENCE RELATION IN NUMBER THEORY Mannu Arya1 and Vipin Verma2 1,2Department of Mathematics, School of Chemical Engineering and Physical Sciences Lovely Professional University, Phagwara 144411, Punjab (INDIA) ABSTRACT The research is on recurrence relation in number theory, with the sole aim of reviewing subtle and far- ending relationships in selected natural numbers which includes the composite and perfect numbers. Related literature was reviewed on which serves as the basis for critical analysis of the natural numbers. From the findings, it was found that there is a degree of relationship between prime numbers and perfect numbers including composite numbers. The nth composite number (푐푛) can be generated using the Wolfram Language code. This is a machine code. It is very difficult to establish a manual formula to extrapolate far composite numbers, but Dirichlet generating function is used to establish the characteristic function of the composite numbers. Furthermore, the relation in perfect numbers can be addressed thus as follows; (Power of Two) × (Double that Power - 1). The prime number formula is given by(2푛−1). To get the Perfect Number, the formula becomes (2푛−1). (2푛 − 1). Getting the nth sequence of a perfect number is dependent on the equivalent nth sequence of prime numbers. The research finally concludes with the call for researchers to team up to make more explicit, the recurrence relation in composite numbers. Keywords: Number Theory, Recurrence Relation, Perfect numbers, composite numbers, wolfram language code, Dirichlet generating function 1.0. INTRODUCTION The study of subtle and far-reaching relationships of numbers is Number Theory [3]. Number Theory is the study of positive numbers (1,2,3,4,5,6,7…) which scrutinize the properties of integers, the natural numbers which is common as -1,-2,0,1,2 and so forth. It is part theoretical and part experimental, as mathematics seeks to discover fascinating and even unexpected mathematical interactions [5]. Since ancient times, people have separated the natural numbers into a variety of different types which includes but not restricted to odd numbers (1,3,5,7,9,11...), cube numbers (1,8,27,64,125...), prime numbers (2,3,5,7,11,13,17,19,23,29,31…), composite numbers (4,6,8,9,10,12,14,15,16…), 1(modulo 4) numbers (1,5,9,13,17,21,25…), 3( modulo 4) numbers (3,7,11,15,19,23,27…), triangular numbers ( 1,3,6,10,15,21…), perfect numbers ( 6,28,496…), Fibonacci numbers ( 1,1,2,3,5,8,13,21….) [4][5]. Many of these types of numbers are undoubtedly already known to us. Others such as the “Modulo 4” numbers may not be familiar. A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4, and similarly for the 3 (modulo 4) numbers. A number is called triangular if the number of pebble can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on. The Fibonacci numbers are created by starting with 1 and 1. Then to get the next number in the list, just add the previous two. Finally, a number is perfect if the sum of all its divisors, other than itself, adds back up to the original number. Thus the numbers dividing 6 are 1, 2 and 3 and 1 + 2+ 3 = 6. Similarly, the divisors of 28 are 1,2,4,7 and 14 and 1+ 2 + 4+ 7+ 14 = 28 [4][5]. Number theory involves analyzing such mathematical relationship, as well as asking new questions about them. The main goal of number theory is to discover interesting and unexpected relationships between different sorts of numbers to prove that these relationships are true. These relationships are drafted using recurrence relation [5]. ISSN: 2005-4238 IJAST 885 Copyright ⓒ 2020 SERSC International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893 Recurrence relation is an equation that defines a sequence based on a rule that gives the next term as a function of the previous terms. The simplest form of a recurrence relation is the case where the next term depends only on the immediate previous term. If we denote the nth term of the sequence by ( Xn). Such a recurrence relation is of the form [2][8][10] Xn+1 = f(Xn) A recurrence relation can also be higher order, where the term Xn+1could depend not only on the previous term Xnbut also on earlier terms such as, Xn−1, Xn−2, etc. A second order recurrence relation depends just onXnand Xn−1and is of the form [2][4][8][10] Xn+1 = f(Xn, Xn−1) For some f with two inputs. For example, the recurrence relation Xn+1 = Xn + Xn−1can generate sequence based on a recurrence relation; one must start with some initial values. For a first order recursion, Xn+1 = f(Xn), one just need to start with an initial value X0 and can generate all remaining terms using the recurrence relation. For a second order recursion, Xn+1 = f(Xn, Xn−1) one needs to begin with two values X0 and X1. Higher order recurrence relations require corresponding more initial values [2][4][8][10]. Amongst the complex natural numbers mentioned earlier, the recurrence relation for Fibonacci, has being highly researched on. So, for the purpose of this research, composite and perfect numbers will be looked into. From the conclusions, observations made on recurrence relation would be used to explain the relationship between the different numbers involved, stating conflicting issues [2][4][8][10]. 2.0. LITERATURE REVIEW This session review related literature on recursion relation of natural numbers. 2.1. Concept of Number Theory Number Theory is the study of positive numbers (1,2,3,4,5,6,7…) which scrutinize the properties of integers, the natural numbers which is common as -1,-2,0,1,2 and so forth. It is part theoretical and part experimental, as mathematics seeks to discover fascinating and even unexpected mathematical interactions. It is the study of subtle and far-reaching relationships of numbers. This far end reaching relationships have its application in computer algorithm as in Fibonacci numbers [4][5]. Since ancient times, people have separated the natural numbers into a variety of different types which includes but not restricted to odd numbers, cube numbers, prime numbers, composite numbers, 1(modulo 4) numbers, 3( modulo 4) numbers, triangular numbers, perfect numbers, Fibonacci numbers, etc. Not withstanding, the major concentration will be on Perfect and composite numbers. Some of these numbers outlined above are summarized in the table below [5]; Table 1: Classifications of Natural Numbers Number Type Samples Odd 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35……………. Cube 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331……………………….. Prime 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 ……………… Composite 4, 6, 8, 9, 10, 12, 14, 15, 16, 17, 19……………………………….. 1 ( Modulo 4) 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49 …………………………. ISSN: 2005-4238 IJAST 886 Copyright ⓒ 2020 SERSC International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893 3 ( Modulo 4) 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59………………. Triangular 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91 ………………………. Perfect 6, 28, 496, 8128, 2096128, 33550336………………………………….. Fibonacci 1, 1, 2, 3, 5, 8, 13, 21 …………………………………………….. Source: Classifications of Natural Numbers [5] The table above has shown some of the common and uncommon natural numbers. Some of the mathematical relationship between some of them can be easily figured out while some are complicated to figure out or have not attained a well-defined explanation of the relationship between them. Some research works however, have shown that there is a relationship between prime and perfectnumbers. For clear and deductive purpose, the composite and perfect numbers will be looked into shortly [5]. 2.2. Recurrence Relation Recursive relation is a recursive definition without the initial conditions. Recurrence relation is an equation that defines a sequence based on a rule that gives the next term as a function of the previous terms. The simplest form of a recurrence relation is the case where the next term depends only on the immediate previous term. If we denote the nth term of the sequence byXnSuch a recurrence relation is of the form Xn+1 = f(Xn)[2], [8],[10],[18], [19]. 2.3. Methods of Recursion Recursion is the process of repetition (repetition), and a recursive method is the use of recursion in a finite or infinite way to define, constructs, or analyzes a mathematical object. Often times a recursive structure will have a closed form counterpart. For instance, the Fibonacci numbers are expressed through the recursive definition Fn = Fn−1 + FN−2forn ≥ 2 such that F0 = 0 andF1 = 1 , with Benit’s formula [2], [8], [10]. ( 1 + √5)푛 − (1 − √5)푛 퐹푛 = 2푛√5 For푛 ∈ 푁 ∈ as the closed form counterpart. The closed form allows for rapid computation of the nth Fibonacci number, but the recursive definition is more convenient for proving identities and a more natural approach to the sequence. Generating functions such as the formal power series ∞ 푛 G (t) = ∑푛=0 푎푛푡 Provide a straightforward connection between the recursively defined anand the closed G (t).
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