A Study on the Usage of Number Theory for the Formulation of Mathematical Problems

Srinivasa Ramanujan has recorded many beautiful continuous fraction extensions in his notebooks. Chapter 12 of Ramanujan 's second notebook is entirely devoted to continuous fractions. Other continuous fractions can be found in other chapters, in particular in his second notebook, Chapter 16[94]. The unorganised portions of the second and third notebooks of Ramanujan contain about 60 additional results for the continued fractions. This part of Ramanujan 's work has been addressed and developed by several authors, including G. E. Andrews, M. D. Carlitz, R. Carlitz Hirschhom L. Carlitz Gordon W. A. Allalam and M. Scoville B. E. H. Ismail and K. G. Ramanathan R. Y. Denis S. Bhargava and Adiga Bhargava from Chandrashekar, Adiga, and D. D. Somashekara Adiga and Somashekara A. Verma, Denis, K. It's Rao Srinivasa and N. The most famous and only continuous fraction shown in Ramanujan's published papers is the continuous fraction of Rogers-Ramanujan and is given by Rogers-Ramanujan. Srinivasa Ramanujan has recorded a few results in his second scratch pad [94, pp. 257-262] in the hypotheses of the elliptical capacity of the elective bases relating to the traditional hypothesis. K. Venkatachaliengar inspected a portion of the sections in the 'lost' note pad of Ramanujan[96] dedicated to elective hypotheses. Ramanujan had a rare appreciation of number n. The vast majority of the hypotheses, in particular the isolated conditions and speculations of the elliptical capacity to the elective bases created by Ramanujan, were devoted to the representational arrangement for 1/;r. So as to set up some wonderful equations for 1/;r expressed by Ramanujan J. M. Borwein and P. B. Borwein has built up "comparing speculation." Ramanujan has recorded 23 delightful P-Q estimated time of arrival work characters, including the remainder of the estimated time of arrival capacity in Chapter 25 of his subsequent scratch pad. A large proportion of these P-Q personalities have been shown by Bemdt and Zhang to use the different measured conditions of Ramanujan having a place with the hypothesis of mark 2 (traditional hypothesis). In order to build up the rest of the P-Q characters, Bemdt used the hypothesis of a particular structure. As of late, Bhargava, Adiga and Mahadeva Naika have acquired another class of P-Q personalities on the basis of certain special conditions having a place with the 'elective hypothesis of mark 4.' In the same way, Baruah[80] has demonstrated five special conditions of Ramanujan. These P-Q personalities are incredibly helpful in calculating class invariants and estimating the proportions of theta-capacity. Ramanujan has recorded a few estimates of (p(q) in his scratch pad[94]. Bemdt and Chan[21] demonstrated each of these qualities and some new estimates (p{q) not guaranteed by Ramanujan on the use of Ramanujan 's measured conditions and class-invariants. In this postulation, we are zeroing in the investigation of some known and new outcomes in the Elementary Number Theory, which examines the arrangement of positive numbers that are additionally called common numbers. From ancient times, mathematicians divided the regular numbers into di erent types of numbers, some of which are as follows[40]: Even numbers: 2; 4; 6; 8;

Composite Numbers: 4; 6; 8; 9; Square numbers: 4; 9; 16; 25; Triangular numbers: 1; 3; 6; 10 Perfect numbers: 6; 28; 496; Fibonacci numbers: 1; 1; 2; 3; 5; 8; There are some standard questions in the theory of numbers like:

Que. Can the total of two squares be a square ? The response to this investigation is an rmative given by Pythagoras (569-470 B.C.) to the western world. Pythagorean hypothesis [44] states that there is a triangle [65] with sides x; y and z, which satisfies the condition x2 + y2 = z2 which is known as Pythagorean condition. The triple x; y and z of positive integer numbers that satisfy the Pythagorean condition is called Pythagorean triples. For example: 32 + 42 = 52; 52 + 122 = 132 etc. The Pythagorean were the rst to give a method of determining in nitely many pyth-gorean triples [chapter 2, theorem (2.1.2)]. About 300 B.C. The presence of Euclid's Elements and the assortment of 13 books has changed over arithmetic from numerology to deductive science. In Book X, he gave a technique to get all the Pythagorean triplets without confirmation. Provides a link between the crude Pythagorean triples and the Mean Value The-orem. Likewise, he talked about the Pythagorean triples whose triangles have a typical hypotenuse, inradius, edge, zone, legsum. Kak and Kothapalli examine the use of crude Pythagorean triplets in cryptography. Roy and Sonia[60] have a di-rect strategy to create all potential crude and non-crude triples for some random number and have developed a method for delivering Pythagorean quadruples and tuples. The Pythagorean Triple Metric Sequences are used to close the connexion. In addition, the Pythagorean Quadraple has been disused. Trios of positive enters. Que. Que. Can we and the trios of positive integer numbers have the ultimate goal that the integer of any two coordinates is some xed power? unmistakable positive numbers with the ultimate goal that any two of them is a square.

The numbers that are organised as some numerical gures, such as tri-edge, square, and so on, are known as gurate numbers. The numbers that look like a triangle are called three-sided num-bers. For example: 1; 3; 6; 10; are three-sided numbers. The numbers that look like squares are called square numbers. Examples are: 1; 4; 9; 16; 25; square numbers. The numbers that are arranged in the shape of a pentagon are called pentagonal num-bers. Examples are: 1; 5; 12; 22; 35; pentagonal numbers. Likewise, the numbers that are arranged in the shape of Polygon are called polygonal numbers. A polygonal number denoted by Pd(n) and is a number of the form Pd(n) = 1 + [1 + (d 2)] + [1 + 2(d 2)] + + [1 + (n 1)(d 2)] Pd(n) is a d-gonal number of order n [nth d-gonal number]. Que: Is there a link between the three-sided number and the other polygonal number-bers? In addition, the response to this investigation is YES. Each polygonal number can be written about three-sided numbers So the tri-precise number assumes a significant function in the investigation of the number hypothesis. The connexion between the Triangular Numbers and the Prime Numbers is described in subtleties. Likewise, it provides clarification of the writing of positive whole number as far as Tri-precise numbers are concerned. From ancient times, three-sided numbers draw attention to individuals all over the world. The intrigue is not limited to mathematicians and scientists. Further some fascinating properties of three-sided numbers can be found in. The nitude of the primes. Que. Que. Are there numerous primes compatible with 1 modulo 4 and congruent with 3 modulo 4? The appropriate response to these enquiries is also an rmative. Indeed, these investigations are the specific instances of the notable Dirichlet hypothesis. The iniquity of the primes is known from the time of Euclid, while the verification of the iniquity of the primes concerning the arithmetic movement is given by Dirichlet after the broken evidence given by Legendre. Dirichlet Theorem: If a; d 2 N and gcd(a; d) = 1, then the nite sequence a + d; a + 2d; a + 3d; contains quite a lot of primes. For proof of theorem, Dirichlet used a complex analysis, so that in the history of Number Theory Dirichlet 's original proof is considered to be non-elementary and an-alytic. From Dirichlet theorem, one can easily conclude that, 1. There are in nitely many prime numbers. 2. There are in nitely many primes of the form 4n + 1 and also of the form 4n + 3, etc.

De nition 1.2.1 (first number). A positive integer p > 1 with no proper posi-tive factor other than 1 is called a prime number. An integer n > 1 that is not prime is called a composite number. The set of primes is countably nite. Theorem 1.2.1 (Fundamental Theorem of Arithmetic). Every integer n > 1 can be written as a product of a premium uniquely, with the prime factors in the product in a non-decreasing order[9]. For integers a; b with 6=0, we say that an is a factor (or divider) of b if b = ka for some k 2 Z, and in this case we write ajb. For integer n > 1 and integers a; b; we say that an is congurent to b modulo n, denoted by b(modulo n) if n is divided by b. This is the equivalence relationship of the set of integers Z[6]. Result 1.2.1. [34] If a b(mod n) and c d(mod n), then 1. a + k b + k(mod n), ak bk(mod n) for any k 2 Z: 2. a c b d(mod n) 3. ac bd(mod n) 4. ak bk(mod n), ak bk(mod nk) for any k 2 N gcd(k; n) = 1, then k k (mod n) 6. If a b(mod ni), 1 i k, then a b(mod N) where N =lcmfn1; n2; ; nkg. In particular a b(mod n1n2) if gcd(n1; n2) = 1, that is n1; n2 are relatively prime, a b(mod n1n2 nk) if n1; n2; ; nk are pairwise relatively prime. (vii) If p is a prime and a2 b2(mod p), then a b(mod p) or a b(mod p): Result 1.2.2. For a; b 2 Z and integer n > 1; let a +n b = the least nonnegative remainder when a + b is divided by n; a n b = the least nonnegative remainder when ab is divided by n: Then for n > 1; U(n) = fk 2 Njk < n and gcd(k; n) = 1g is an abelian group under n with identity 1 and its order is (n); where is called Euler‘s phi function. For any a 2 Z; with gcd(a; n) = 1 we have r 2 U(n) such that a r(mod n): Then we de ne o(a) as o(a) = o(r) = k is the least positive integer such that rk = 1 (i.e ak rk 1(mod n)). Clearly ao(a) 1(mod n) and o(a)j (n): Note that is multiplicative and (pk) = pk pk 1 for a prime p and k 2 N: A positive integer n is prime i (n) = n 1: For a prime p, a is its own inverse in group U(p) I a 1(mod p) or a 1(mod p): Theorem 1.2.2 (Wilson‘s Theorem). [47] An integer n > 1 is prime i (n 1)! 1(mod n): Theorem 1.2.3 (Fermat‘s Little Theorem (FLT)). [47] If p is a prime, then ap a(mod p) for all integers a: If gcd(a; p) = 1, then ap 1 1(mod p). Converse of FLT is not true for example, 2341 2(mod 341) and 341 = 11 31 which is not prime. Theorem 1.2.4 (Euler‘s Theorem). [40] If integer n > 1 and gcd(a; n) = 1, then 1(mod n): Remark 1.2.1 (Contrapositive of FLT (Primality test)). For integer n > 1; if there exist a 2 N with an 6 a(mod n), then n is a composite number. Example 1.2.1. As 263 = 260 23 = (26)10 8 = (64)10 8 110 8 8(mod 63) i.e. 263 6 2(mod 63), so 63 is not prime. OR As gcd(2; 63) = 1 and 262 4(mod 63); i.e 262 6 1(mod 63), i.e. (63) 6 62; so 63 is not a prime.

of r simultaneous linear congruences has a unique solution (mod N): De nition 1.2.2 (Legendre Symbol and Jacobi Symbol ). [39] Let p be an odd prime, a 2 Z and p - a: If x2 a(mod p) has solution, then we say that a is a quadratic residue (mod p) , otherwise a quadratic nonresidue (mod p). There are 12 (p 1) quadratic residues and equally many quadratic nonresidues (mod p). Legendre symbol (ap ) is de ned as : Properties:[39] ] Let a, b ∈ Z such that p - ab (p an odd prime). Then mathematics about the great Indian mathematician Srinivasan Ramanujan. In Cambridge, while working with Hardy (1913), Ramanujan was ill and admitted to Putney Hospital. Hardy came to visit him in a taxi, the number of which was very dull for him in 1729. Actually, Ramanujan found that number to be very interesting. He said that 1729 was the smallest num-ber which could be expressed in two d erent ways as a sum of two positive cubes. As a result, the numbers which are the taxicab numbers are de ned as those for which solutions are found in the positive integers of the equation, m = a3 + b3 = c3 + d3 where fa; bg = 6 fc; dg. The nth taxicab number is a positive integer which can be expressed as a sum of two cubes of positive integers in n di erent ways. The smallest number of the taxi is Ta(n). The concept of the second taxi number was first mentioned in 1657 by Bernard Freni-cle de Bersy and was made famous in the early 20th century by the storey of Srinivasa Ramanujan and G. H. Hardy, man. G. In 1938. Uh, H. Hardy and E. M. Wright has shown that such numbers exist for all positive integers, and that their proof is easily converted into a programme to generate such numbers. However, the proof does not state at all whether the numbers thus generated are the smallest positive and therefore cannot be used for the actual value of Ta(n).Following six taxicab (smallest in size) are known .

Ta(1) = 2 = 13 + 13 Ta(2) = 1729 = 13 + 123 = 93 + 103 Ta(3) = 87539319 = 1673 + 4363 = 2283 + 4233 = 2553 + 4143 Ta(4) = 6963472309248 = 24213 + 190833 = 54363 + 189483 = 102003 + 180723 = 133223 + 166303 Ta(5) = 48988659276962496 = 387873 + 3657573 = 1078393 + 3627533 = 2052923 + 3429523 = 2214243 + 3365883 = 2315183 + 3319543 Ta(6) = 24153319581254312065344 = 5821623 +289062063 = 30641733 +288948033 = 85192813+286574873 = 162180683+270932083 = 174924963+265904523 = 182899223+ 262243663

Ta(2) is also known as the Hardy-Ramanujan number. The subsequent taxicab num-bers were found with the help of supercomputers. John Leech obtained Ta(3) in 1957. Ta(4) found in 1991 [59]. J. A. Dardis found Ta(5) in 1994 and it was con rmed by David W. Wilson [19] in 1999. In 2003 Calude S.etal found Ta(6) [10]. Ta(6) was announced by Uwe Hollerbach on March 9, 2008. In [14] upper bound for taxicab and cabtaxi numbers are given. Cubefree number means a positive integer that is not divisible by any p3 where p is a prime. If a cubefree taxicab number T is written as T = x3 + y3, then x and y are relatively prime. Among the taxicab numbers Ta(n); 1 n 6; only Ta(1) and Ta(2) are taxicab number with three representations was discovered by Paul Vojta in 1981 while he was a graduate student. It is The smallest cubefree taxicab number with four representations was discovered by Staurt Gascoigne and independently by Duncon Moore in 2003. It is 1801049058342701083 = 922273 + 12165003 = 1366353 + 12161023 = 3419953 + 12076023 = 6002593 + 11658843. Positive integers, three representations, not cube free

Chapter 2. Pythagorean Triples Section 2 examinations the di erent properties of crude Pythagorean triples with its applications to di erent zones of Mathematics. We likewise demonstrated some new properties of Pythagorean triples. We have talked about certain nuts and bolts of compatible numbers. We have additionally talked about 1) Existence of Pythagorean triples as far as A. P. what's more, nonexistence in G. P. 2) Nonexistence of Pythagorean triples regarding Harmonic Progressions 3) Irrationality of some genuine numbers like √ 2, √ 3, √ 5, and so on with the assistance of various properties of Pythagorean triples. For uses of Pythagorean triples one may utilize and properties of Pythagorean triples. For utilizations of Pythagorean triples one may utilize and. we have found di erent kinds of triples of positive numbers with the end goal that the total of any two directions is an ideal squares, 3D shapes, fourth powers, fth powers and so forth. We have examined the accompanying outcomes in subtleties in this part. For the instance of impeccable squares use R1: For assurance of unmistakable a; b; c 2 N with n 2, to such an extent that a + b; a + c; b + c are nth intensity of positive numbers. positive numbers with the end goal that ja bj; ja cj; jb cj are immaculate squares. These sorts of triples are emerges from Pythagorean triples which we talked about in section 2. R3: Determination of a trio (a; b; c) of particular positive whole numbers with the end goal that a+b; a+ c; b + c are flawless squares. R4: Triplets with the assistance of a known trio. R5: Triplets from four tuples with a similar property. With the assistance of the conditions we have chosen s; t interms of r and found the triplets with sum of any two coordi-nates is cube, fourth power of positive integers. We have generalised this result for given any positive integer n. R6: Four tuples of distinct positive integers such that sum of any two of its coordi-nates is cube of a positive integer. Also with the help of Taxicab numbers we have found triplets with sum of any two coordinates as cube of a positive integer. There are many different techniques for finding the value of Xn r=1 g(r) where g(r) is a polynomial in r[30]. We've developed a very simple technique to find the value of Xn r=1 g(r). Knowing the result for higher degrees, we have achieved results for lower degrees through a process like differentiation. These results are also obtained through Newton's forward difference interpolation formula and through the use of difference operator operation. In this chapter, we have developed various techniques for the study of the finite sum of polynomial expressions, which are: Integral Technique which a) determines Pn p for p = 1, 2, 3, · · · and Degrees by a process like di erentiation. These results are also obtained by Newton‘s forward di erence interpolation formula and by using di erence operator operation. In this chapter we have

1) Integral Technique which a) determines P np for p = 1; 2; 3; and b) Determines Pg(n) when g(n) is a polynomial. 2) Differentiation Techniques to determine Pn p , p ≥ 0. 3) Forward Difference Techniques With the help of these we have proved the following formulae Triangular numbers are gurate numbers as they are represented by some xed geo-metric patterns.

Proof. If 2n−1 n+7 is an integer, then n + 7|2n − 1. Utilizing Theorem 1.4, we have that n + 7|2(n + 7). Using a comparable speculation again, this time with derivation we have n + 7|2(n + 7) − (2n − 1) which in the wake of rearranging implies n + 7|15. All the components of 15 are −15, −5, −3, −1, 1, 3, 5, 15 and these are for the most part the qualities that n + 7 can take, so the arrangements are: n = −22, −12, −10, −8, −6, −4, −2, 8.

interest in joining number theory and geometry. 2. To study Arithmetic of algebraic objects is indeed very close to that of natural numbers.

The proposed strategies examine the added substance and the multiplicative properties of the number-growth capability. Number-crunching capacity includes a true or complex valued capacity a(n) to determine the normal number set that expresses some math property of n. Added substance and multiplication capacity is the most significant amount of juggling capacity. Added substance work is the math set f(n) of + ve number n, when a, b is co-prime, the output capacity is the capacity option. This capacity f (n) is considered to be a completely added substance if f (a+b) = f (a) + f (b) + ve numbers and b are not co-prime. Completely added substance is to be used in this case by simple use with absolutely multiplicative functions. At the point where f is a completely added substance, work f (1) = 0. The audit of ongoing writing on the added substance and the multiplication capacity is discussed. The properties of the two capacities are examined in detail. Adjusted binomial ring elements for added substance and multiplication capacity are proposed. The ring structure over the mixture of the added substance and the multiplicative capacity can be performed with the modified binomial capacity. The Ring is a set of R, formed with double parallel tasks of expansion and duplication, correspondingly spoken to expansion and deduction. Fundamentally, R is also an abelian set, with zero as the character. Clearly, in the ring, it is plausible to extend, deduct and increase without isolating the set, anyway in the slant field, division can also be proceeded. Further, Isomorphism and Ring Homomorphisms are inferred in this hypothesis. Added capacity of the substance in aggregate primes is the set of multiplication capacity squares. The fundamental hypothesis of math is that every n > 1 can be connoted as a result of unmistakable prime elements. Added substance Prime number Theory was concerned about the delineation of whole numbers as aggregates of primes or of closely related whole numbers. The premiums are described as multiplicative properties, while the issue includes the properties of the added substance. Each + ve number can only be specified as a different prime number. In this strategy, prime number hypothesis, added substance issue including prime, binomial infringement point law for added substance prime markers, prime component hypothesis for added substance arrangement and finally additive works are described. In addition, the and the multiplication capacity are further investigated. This hypothesis manages dynamic strategies with an invariant measure and related issues. It is used for different segments of arithmetic which, in large part, incorporates the development of periodicity properties for unrivalled type techniques. Another critical part of the hypothetical ergodic idea is the question of the metric application of strategies. A fantastic part of this hypothesis and its use of stochastic techniques is used by different ideas of entropy for dynamic strategies. In addition, the portrayal of ergodicity has broken down. At that point, the meanings of the ergodic hypothesis are discussed. In addition, the ergodic hypothesis, including algorithmically arbitrary grouping, is depicted. In the long run, the modified ergodic hypotheses are determined. The grouping of the added substance and the multiplication capacity, characterised at an arbitrary stage, is inspected. In an irregular stage, Rn is the symmetrical gathering and enveloping of all possible capacities that bijectively map the set of first n numbers {1,2, ......,} n into itself. What could be compared to Kolmogorov-Rogozin imbalance is used to decide on the centralization of totally added substance work characterised by irregular stages. For the multiplicative capacity, the Voronoi summability is used to dissect the value of the multiplicative capacity grouping on irregular change.

1. C. Adiga (1992). On the representations of an integer as a sum of two or four triangular numbers, Nihonkai Mathematical Journal, vol. 3, No.2, pp. 125-131. 2. C. Adiga, B. C. Bemdt, S. Bhargava and G. N. Watson (1985). Chapter 16 of Ramanujan 's second notebook: Theta-functions and q-series, Mem. Amer. Math. Soc, No. 315, 53, Amer. Math. Soc, Providence, 1985. 3. C. Adiga, R.Y. Denis and K.R. Vasuki (2001). On some continued fraction expansions for the ratios of 2W 2, Special Functions: Selected Articles, P. K. Baneiji (Ed.), Scientific Publishers (India), pp. 1-16. 4. C. Adiga, M. S. Mahadeva Naika and Ramya Rao (2002). Integral representations and some explicit evaluations of a continuedfraction of Ramanujan, JP Jour. Algebra, Number Theory & Appl. 2(1), pp. 5-20. 5. C. Adiga, M. S. Mahadeva Naika and K. Shivashankara (2002). On some P-Q etafunction identities of Ramanujan, Indian J. Math., vol. 44, No. 3, pp. 253-267. some Rogers-Ramanujan type continued fraction identities, Mathematical Balkanica, New Series, vol. 12N0S. 1-2, pp. 37-45. 7. C. Adiga, Taekyun Kim, M. S. Mahadeva Naika and H. S. Madhusudhan, On Ramanujan's cubic continued fraction and explicit evaluations of theta-functions, Indian J. Pure and Applied Mathematics, (to appear). 8. C. Adiga, K. R. Vasuki and M. S. Mahadeva Naika (2001). On some new identities involving integrals of theta-functions, Advanced Studies in Contemporary Mathematics, 3, No. 2, pp. 1-11. 9. C. Adiga, K. R. Vasuki and M. S. Mahadeva Naika (2002). Some new explicit evaluations of Ramanujan's cubic continued fraction, New Zealand Journal of Mathematics, 31, pp. 1-6. 10. W. A. Al-Salam and M. E. H. Ismail (1983). Orthogonal polynomials associated with the Rogers-Ramanujan continued fraction, Pacific J. Math., 104, pp. 269-283.

Research Scholar (Part Time) OPJS University, Churu, Rajasthan