Recurrence Relation in Number Theory

Home , Composite number, List of prime numbers, Powerful number, Sphenic number

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].

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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 ………………………….

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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). Interplay between a recursive definition and its closed form can be a powerful tool, but many modern problems lack either the recursive relation or the closed form or their definitions are impractical for computational purposes. Constructing a set from other sets is also another common method of recursion. Consider the following case. Given initial set퐾1 = {11} , the rule to obtain퐾푖+1is to first add a 1 to all the integers in all the words of퐾푖. Second, form 2n−1 new words by placing exactly one (1) in all possible positions between the integer in each word. To finally obtain퐾푖+1, concatenate a 1 to the front of each word. The word 11

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International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893

becomes 22 whichgenerate 122, 212, and 221, leading to the new set 퐾2= {1122, 1212, and 1221}. Continuing in this Fashion(n − 2) more times will yield a set of all double occurrence words with n symbols. It is clear from this construction that |퐾푛+1| >|퐾푛| for all n ∈N. By contrast, the cardinality of the set formed by taking the cardinality of the set formed by takingsuccessive differences of {1, 4, . . . 1 ∞ n2}decreases with each iteration. However, starting from an infinite set( ) and taking successive 푛 푛=1 differences will not change the cardinality [2], [4], [8], [10]. Set constructions that increase in cardinality are common in enumeration problems, and give an upper bound to questions of algorithmic complexity and asymptotes. By contrast, set constructions which decrease the cardinality are useful in sieving and optimization problems, where a solution or set of solutions are refined with each iteration

3.0. RESEARCH METHODOLOGY AND DESIGN Research methodology refers to the steps, procedures and strategies used for gathering data in the course of the research investigation (Polit and Hungler, 1997). This chapter is concerned with methodology used in achieving the research aim. It also contains the focus the of study and instrument of data collection used. 3.1. Focus of the Study The study focuses its intention on recurrence relation in perfect and composite numbers. It is hinged at analyzing the mathematical relationships between them, as well as asking new questions about them and to prove that these relationships are true.

Natural Numbers of concentration; Perfect and Composite Numbers 3.2. Sources of Data Secondary source of data was the major source of data. This was where the major source of information was gathered. The researcher consulted published and unpublished books, journals, and web. 3.3. Research Design A research design according to Andrew Kirumbi as cited in Muaz (2013) is the set of methods and procedures used in collecting and analyzing measures of variables specified in the research problem research. The research paper adopts the pure research method as it considers figuring out the recurrence relation in natural numbers. 3.4. Recurrence Relation in Composite and perfect Numbers 3.4.1. Composite Numbers A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite or the unit 1. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself [13], [17]. The composite numbers up to 150 are; 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150. (Sequence A002808 in the OEIS as cited in [1] [13] [17]) Every composite number can be written as the product of two or more (not necessarily distinct) prime numbers. For example, the composite number 299 can be written as 13 × 23, and the composite number

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International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893

360 can be written as 23 × 32 × 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic [1] [13] There are several known primality tests that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input. However, one of the ways to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semi-prime or 2-almost prime (the factors need not be distinct; hence squares of primes are included. A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For the latter[1] [13][17] If all the prime factors of a number are repeated it is called a powerful number (All perfect powers are powerful numbers). If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 23 × 32, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 are squarefree.

Table 2: Table of List of some composite numbers with their prime factorization

Composite numbers ( Prime factorization Composite numbers Prime factorization n) ( n) 4 22 15 3 x 5

6 2 x 3 16 24

8 23 18 2 x 32

9 32 20 22 x 5

10 2 x 5 21 3 x 7

12 22 x 3 22 2x 11

14 2 x 7 24 23 x 3 Source: Table of List of some composite numbers with their prime factorization[1] [13] [17] Composite numbers can also been called "rectangular numbers", but that name can also refer to the pronic numbers, numbers that are the product of two consecutive integers. Yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed (prime) number. Such numbers are called smooth numbers and rough numbers, respectively[1], [12]. . The first few composite numbers ( sometimes called “ composites” for short) are 4,6,8,9,10,12, 14,15,16, (OEIS A002808), whose prime decompositions are summarized in the following table. It was also noted that the number 1 is a special case which is considered to be neither composite nor prime. It is worth to note that the nth composite number (푐푛) can be generated using the Wolfram Language code, which allows programmers to operate at a significantly higher level than ever before, by leveraging built-in computational intelligence that relies on a vast depth of algorithms and real-world knowledge

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International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893

carefully integrated over three decades. However, the Dirichlet generating function of the characteristic function of the composite numbers is given by [12] [13] ;

∞ 1 ∑ 푠 퐶푛 푛=1

1 1 1 1 = + + + + ⋯ 4푠 6푠 8푠 9푠 (푠)-1 – P (s) ﻡ= .s) is the Riemann Zeta function, p (s) is the prime Zeta function, and (s) is an iverson bracket)ﻡ Where The composite number problem asks if there exist positive integers m, and n such that N= m x n. The recurrence relation in composite has being very difficult to figure, but however, some distinct relationship was observed. The first 20 sequence of the composite has its common differences ( nth term - (n-1)term ) as 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1,2, 2. The prime numbers are the key factors to the formation of the composite numbers [1] [17] . It follows a chronological arrangement of natural counting numbers, but with only manipulation of all prime numbers. Example, From the prime numbers; 2, 3,5,7,11,13,17,19,23,29,31, etc. Following natural counting number inline with the formula for composite numbers ( m x n) where m and n are all prime numbers, the first term will be 2 x 2 = 4, next 2 x 3 =6, 3 x 3 =9, etc. The manipulation is such that no number is left in between, example one may not consider 3 x 3, and directly go to multiplying 2 by 5 , to get 10. However, this is however laborious getting to far ending composite numbers, so, the nth composite number (cn) can be generated using the Wolfram Language code as mentioned above [7][17].

3.4.2. Perfect Numbers A perfect number is a whole number, an integer greater than zero; andwhen you add up all of the factors less than that number, you get that number. Perfect numbers are stemmed from realizing numbers whose divisors adds up to it self. Starting with the number 1, it is not a perfect number because it can be divided by only itself, 2, 3, 4, 5, are also not perfect numbers. The closest perfect number is 6 [9][14][15]. Examples: The factors of 6 are 1, 2, 3 and 6. 1 + 2 + 3 = 6 The factors of 28 are 1, 2, 4, 7, 14 and 28. 1 + 2 + 4 + 7 + 14 = 28 The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496. 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128 1+ 2+ 4+ 8+ 16+32+ 64+ 127+ 254+ 508+ 1016+ 2032+ 4064 = 8128 According to the Merriam-Webster Dictionary, the term was first used in the fourteenth century. The Grolier Multimedia Encyclopedia says that perfect numbers are "another example of Greek progress in number theory," and credits the Pythagoreans for coining the term "perfect. [16]" The first four perfect numbers were known over 2,000 years ago. Some ancient cultures gave mystic interpretations to numbers that they thought were magic [9][15].

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International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893

It is noticed that every perfect number have a higher power of two than the prior number up to a certain number, and then a prime number that is equal to DOUBLE the last power of two, minus one (1) [9][14][16]. 1, 2; (2 ×2) -1 = 3 1, 2, 4; (4 ×2) -1 = 7 1, 2, 4, 8, 16; (16×2) -1 = 31 1, 2, 4, 8, 16, 32, 64; (64×2) –1=127 1,2,4,8,16,32,64,128,256,512,1024,2048,4096; ( 4096×2) -1 = 8191 The rest of the factors are each power of two TIMES that prime number.So, our first five perfect numbers are; 2× 3 = 6; 4 ×7 = 28; 16 ×31 = 496; 64 * 127 = 8,128; 4096 × 8191 = 33,550,336 The relation can be addressed thus as follows; (Power of Two) × (Double that Power - 1) The prime number formula is given by (2n-1). In the above examples, we would have (22-1), (23-1), (25-1), (27-1), and (213-1). This would make the other number 2n-1, or 21, 22, 24, 26, and 212. To get the Perfect Number, the formula becomes (2n - 1)x(2 n -1) [3][9][[15] Where "n" is one of a very short list of prime numbers that can be used to create "Mersenne" prime numbers; The illustration is thus as follows; 21 x (22 - 1) = 2 x 3 = 6 22 x (23 - 1) = 4 x 7 = 28 24 x (25 - 1) = 16 x 31 = 496 26x (27 - 1) = 64 x 127 = 8,128 212 (213-1) = 4096 x 8191 = 33,550,336 216 x (217-1) = 65536 x 131071 = 8,589,869,056 218 x (219-1) = 262144 x 524287 = 137,438,691,328 230 x (231-1) = 1073741824 x 2147483647=2,305,843,008,139,952,128 So far, according to the Mersenne organization, there are 37 known Mersenne prime numbers. This means that there are 37 known "perfect" numbers. The mersenne prime number was discovered on January 27, 1998 [12] [16]. Obviously, this is not all of them. It is very likely; too, that there are many more that we will never know. The first term of the perfect number is gotten by substituting “n” with the first term of prime numbers. The first 35 perfect numbers fit this same formula with "n" values of:2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, and 1398269 [ 3][9]. The first eight are calculated and shown above. Another perfect number that has been found is equal to 22976220 x (22976221-1) [3][9][12]. If this number is multiplied out, the perfect number gotten will be 1,791,864 digits long. It is not clear is this would be the 36th, 37th, 38th or nth perfect number. However to find the position of the perfect number, the position of the prime number needs to known as it is crystal clear that perfect numbers depends on prime numbers [9][15][16].

4.0. FINDINGS AND CONCLUSIONS The research finds out that there is a degree of relationship between prime numbers and perfect numbers including composite numbers. In composite numbers, using the prime numbers; 2, 3,5,7,11,13,17,19,23,29,31, etc. and following natural counting number inline with the formula for composite numbers ( m x n) where m and n are all prime numbers, the first term will be 2 x 2 = 4, next 2 x 3 =6, 3 x 3 =9, etc. The manipulation is such that no number is left in between, example one may not consider 3 x 3, and directly go to multiplying 2 by 5 , to get 10. However, this is however laborious getting to far ending composite numbers, so, the nth composite number (푐푛) can be generated using the

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International Journal of Advanced Science and Technology Vol. 29, No.02, (2020), pp. 885-893

Wolfram Language code as mentioned above. 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 (2n-1). To get the Perfect Number, the formula becomes (2n - 1)x(2 n -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.

ACKNOWLEDGEMENT The regards to this research first Go to God almighty for the grace he has given unto me to achieve this research work. The financial support was solely from the researcher with little aid from my friends, which also further backed up by advices from colleagues and top professors of the department of Mathematics.

LIST OF TABLES Table 1: Classifications of Natural Numbers

Table 2: List of some composite numbers with their prime factorization

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