DOCSLIB.ORG

Forms and Methods of Solutions of Indeterminate Equations of Orders Greater Than One by Ancient Indian Mathematicians

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Forms and Methods of Solutions of Indeterminate Equations of Orders Greater Than One by Ancient Indian Mathematicians See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/256088117 Forms and methods of solutions of indeterminate equations of orders greater than one by indian mathematicians Article · August 2013 CITATIONS READS 0 737 1 author: Vivek Sinha University of Amsterdam 11 PUBLICATIONS 70 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: research View project All content following this page was uploaded by Vivek Sinha on 16 May 2014. The user has requested enhancement of the downloaded file. Forms and methods of solutions of indeterminate equations of orders greater than one by Ancient Indian mathematicians Vivek Sinha (09 MS 066) IISER Kolkata In this paper the forms and methods of solutions of indeterminate equations of orders greater than one by some eminent ancient Indian mathematicians like Bhaskaracharya (II), Brahmagupta, Nrayana Pandit etc. have been discussed with highlighting examples. Special focus has been given to the cyclic (chakravala) method for solving the Diophantine equation of second degree viz. Nx2 + K = y2. “Time and again world has looked up upon us such that we know nothing, and of course, we gave them nothing ( the zer0)!” Anonymous 1. INTRODUCTION Indeterminate equations of orders greater than one fall under the special category of equations called Diophantine equations. *Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. Examples of Diophantine equations* In the following Diophantine equations, x, y, and z are the unknowns, the other letters being given are constants. This is a linear Diophantine equation For n = 2 there are infinitely many solutions (x,y,z), the Pythagorean triples. For larger values of n, Fermat’s last theorem states that no positive integer solutions (x, y, z) satisfying the equation exist. (Pell’s equations) which is named after the English mathematician John Pell. It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century. * http://en.wikipedia.org/wiki/Diophantine_equation India's contribution to integral solutions of Diophantine equations can be traced back to the Sulbha Sutras, which were Indian mathematical texts written between 800 BC and 500 BC. Baudhayana (circa 800 BC) found two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also attempted to solve simultaneous Diophantine equations with up to four unknowns. Apastamba(circa 600 BC) attempted simultaneous Diophantine equations with up to five unknowns. Diophantine equations were later extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for determination of integral solutions of Diophantine equations. Systematic methods for finding integer solutions of Diophantine equations could be found in Indian texts from the time of Aryabhata AD (499). Brahmagupta (628) handled more difficult Diophantine equations - he investigated Pell's equation, and in his Samasabhavana he laid out a procedure to solve Diophantine equations of the second order, such as 61x2 + 1 = y2. It is interesting to note that these methods were unknown in the west, and in fact this very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat. It was solved seventy years later by Euler. However, the solution to this equation was found more than centuries ago by Bhaskara II in his book Lilavati (1150) (who also found the solution to Pell's equation), using a modified version of Brahmagupta's method(Brahma Sphuta Siddhānta). Bhaskara II, in his Bijaganita (Algebra), recognized that a positive number has both a positive and negative square root, and solved quadratic equations with more than one unknown, various cubic, quartic and higher-order polynomial equations, Pell's equation, the general indeterminate quadratic equation, as well as indeterminate cubic, quartic and higher-order equations. It is also amazing to know that the Pell’s equation was never solved by Pell. In fact this equation should rightly be called as Brahamgupta’s equation. We shall now discuss the cyclic method of solving the indeterminate quadratic 2 2 equation, Nx + K = y . 2.CYCLIC METHOD FOR SOLUTION OF THE INDETERMINATE EQUATION, 2 2 Nx +k = y 2.1Brahmagupta’s lemma: 2 2 If (s,t) is a positive integral solution(found by trial) to the equation Nx +k = y 2 2 2 2 2 2 2 then (2st,Ns + t ) is a solution to Nx +k = y , or (2st/k,(Ns + t )/k) is a solution to Nx2+1 = y2. Here N, s, t >0 and are integers. Bhaskara’s improvement: The cyclic method improves the Brahmagupta’s method. Here, we don’t have to look for one solution by trial but we can get it by calculation. The proof of Brahamgupta’s lemma and Bhaskara II’s further work has been given in supplementary material. We shall now see an example on application of Brahmagupta’s Samāsa operation. EXAMPLE 1 Consider the equation 3x2+1 = y2. On comparing with the standard form of the equation Nx2+k = y2, we have N = 3 and k = 1. By observation, we find one integral solution (1,2). Applying Samāsa operation between the solution (1,2) with itself we get that (4,7) is also a solution. Now applying the same operation between (1,2) and (4,7) we get that (15,26) is also a solution. Continuing this way we get that(1,2),(4,7),(15,26),56,97),(209,362)…….. (infinite number of sets). Now we will see an example based on the Cakravāla method. EXAMPLE 2 Let us consider the equation 61x2+1= y2. (1,8) is an integral solution to the equation 61x2+k= y2 with k = 3. In the identity given in Bhaskara’s work, we choose m so that (m+8)/3 is an integer ams |m2-61| is least. We get m = 7, and for this value of m, the identity becomes 61*5*5 – 4 = 39*39. Thus (5/2,39/2) is a solution of the equation 61x2-1= y2, and this is in a standard form. Using Brahmagupta’s Samāsa operation , we find that (195/2,1523/2) is a solution of the equation 61x2+1= y2. Now applying Samāsa operation between (5/2,39/2) and (195/2,1523/2) we get (3805,29718). Repeating the process between (3805,29718) and itself gives (226153980,1766319049) as a solution. Narayan Pandit, another eminent ancient Indian mathematician is also known to work on indeterminate equation s of order higher than one. Following is an example taken from his book Bijganita Vatamsa written in 1350 A.D. EXAMPLE 3 Consider the equation 97x2+1 = y2. Here we see that (1,10) is an integral solution of 97x2+k = y2, with k = 3. In Bhaskar’s identity, mis to be so chosen that (m+10)/3 is an integer and |m2-97| is least. This gives m = 11, and for this value of m, the identity becomes 97*(7)2+8 = (69)2. Thus (7,69) is an integral solution of 97x2+8 = (69)2. For the latter equation, u = 7, v = 69,k= 8 and N = 67. m is now to be chosen such that (7m+69)/8 is an integer and |m2-97| is least, thus giving m = 13, and for this value of m, the identity we have is 97*(20)2+9 = (197)2. Thus (20,197) is an integral solution of 97x2+9=y2. Now for this equation u = 20,v = 197 and k = 9. So now we choose m such that (20m+197)/9 is an integer and that |m2-97| is least. We can see that m = 5. So the identity becomes 97*(33)2+8 = (325)2, showing that (33,325) is an integral solution of 97x2+3 = y2. For this equation u = 33, v= 325 and k = 8. So now we choose m in such a fashion that (33m+325) /8 is an integer and |m2- 97| is least. We have m = 11, and for this the identity that we have is 97*(86)2+3 = 8472. Thus the new equation that we have Is 97x2+3 = y2 with (86,847) as an integral solution. Now for this equation u = 86, v= 847 and k = 3. So now we choose m such that (86m+847)/3 is an integer and |m2-97| is least. We see that m = 10 satisfies such conditions. So the Bhaskara’s identity becomes 97x2+1 = y2. But this equation is in a standard form and its solution therefore can be obtained as [x,y] = [(2*569*5604)/1,(97*(569)2+(5604)2)/1] Thus x = 6377352,y = 62809633 by applying Samāsa method. 3. DISCUSSION If we examine the methods used to solve the indeterminate equations closely we can marvel at the genius of these classic mathematicians. One can see the reflection of an intuition guided marvellous and extraordinarily simple solution to apparently complex problems. The algorithm used for solving the Diophantine equations is something similar to what one uses in modern day programming. As we see in example no. 2, mere verification of the solution is not easy, even if one uses a normal calculator! No wonder that people using methods developed by ancient Indians for basic mathematical operations can calculate really fast (even faster than a computer some times!). Highly intellectual and given to abstract thinking as they were, one would expect the ancient Indians to excel in mathematics$.
Load more

Recommended publications
  • Rationale of the Chakravala Process of Jayadeva and Bhaskara Ii
    HISTORIA MATHEMATICA 2 (1975) , 167-184 RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II BY CLAS-OLOF SELENIUS UNIVERSITY OF UPPSALA SUMMARIES The old Indian chakravala method for solving the Bhaskara-Pell equation or varga-prakrti x 2- Dy 2 = 1 is investigated and explained in detail. Previous mis- conceptions are corrected, for example that chakravgla, the short cut method bhavana included, corresponds to the quick-method of Fermat. The chakravala process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, etc.) are correctly understood only with the new theory. Den fornindiska metoden cakravala att l~sa Bhaskara- Pell-ekvationen eller varga-prakrti x 2 - Dy 2 = 1 detaljunders~ks och f~rklaras h~r. Tidigare missuppfatt- 0 ningar r~ttas, sasom att cakravala, genv~gsmetoden bhavana inbegripen, motsvarade Fermats snabbmetod. Cakravalaprocessen motsvarar en halvregelbunden b~st- approximerande algoritm av minimal l~ngd med flera djupt liggande minimeringsegenskaper. De periodvis upptr~dande storheterna (jyestha-m~la, kanistha-mula, ksepaka, kuttakara, os~) blir forstaellga0. 0 . f~rst genom den nya teorin. Die alte indische Methode cakrav~la zur Lbsung der Bhaskara-Pell-Gleichung oder varga-prakrti x 2 - Dy 2 = 1 wird hier im einzelnen untersucht und erkl~rt. Fr~here Missverst~ndnisse werden aufgekl~rt, z.B. dass cakrav~la, einschliesslich der Richtwegmethode bhavana, der Fermat- schen Schnellmethode entspreche. Der cakravala-Prozess entspricht einem halbregelm~ssigen bestapproximierenden Algorithmus von minimaler L~nge und mit mehreren tief- liegenden Minimierungseigenschaften. Die periodisch auftretenden Quantit~ten (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, usw.) werden erst durch die neue Theorie verst~ndlich.
    [Show full text]
  • IJR-1, Mathematics for All ... Syed Samsul Alam
    January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity.
    [Show full text]
  • Quantum Algorithm for Solving Pell's Equation
    CS682 Project Report Hallgren's Efficient Quantum Algorithm for Solving Pell's Equation by Ashish Dwivedi (17111261) under the guidance of Prof Rajat Mittal November 15, 2017 Abstract In this project we aim to study an excellent result on the quantum computation model. Hallgren in 2002 [Hal07] showed that a seemingly difficult problem in classical computa- tion model, solving Pell's equation, is efficiently solvable in quantum computation model. This project explains the idea of Hallgren's quantum algorithm to solve Pell's equation. Contents 0.1 Introduction..................................1 0.2 Background..................................1 0.3 Hallgren's periodic function.........................3 0.4 Quantum Algorithm to find irrational period on R .............4 0.4.1 Discretization of the periodic function...............4 0.4.2 Quantum Algorithm.........................5 0.4.3 Classical Post Processing.......................7 0.5 Summary...................................8 0.1 Introduction In this project we study one of those problems which have no known efficient solution on the classical computation model and are assumed hard to be solved efficiently, but in recent years efficient quantum algorithms have been found for them. One of these problems is the problem of finding solution to Pell's equation. Pell's equation are equations of the form x2 − dy2 = 1, where d is some given non- square positive integer and x and y are indeterminate. Clearly, (1; 0) is a solution of this equation which are called trivial solutions. We want some non-trivial integer solutions satisfying these equations. There is no efficient algorithm known for solving such equations on classical compu- tation model. But in 2002, Sean Hallgren [Hal07] gave an algorithm on quantum compu- tation model which takes time only polynomial in input size (log d).
    [Show full text]
  • Equation Solving in Indian Mathematics
    U.U.D.M. Project Report 2018:27 Equation Solving in Indian Mathematics Rania Al Homsi Examensarbete i matematik, 15 hp Handledare: Veronica Crispin Quinonez Examinator: Martin Herschend Juni 2018 Department of Mathematics Uppsala University Equation Solving in Indian Mathematics Rania Al Homsi “We owe a lot to the ancient Indians teaching us how to count. Without which most modern scientific discoveries would have been impossible” Albert Einstein Sammanfattning Matematik i antika och medeltida Indien har påverkat utvecklingen av modern matematik signifi- kant. Vissa människor vet de matematiska prestationer som har sitt urspring i Indien och har haft djupgående inverkan på matematiska världen, medan andra gör det inte. Ekvationer var ett av de områden som indiska lärda var mycket intresserade av. Vad är de viktigaste indiska bidrag i mate- matik? Hur kunde de indiska matematikerna lösa matematiska problem samt ekvationer? Indiska matematiker uppfann geniala metoder för att hitta lösningar för ekvationer av första graden med en eller flera okända. De studerade också ekvationer av andra graden och hittade heltalslösningar för dem. Denna uppsats presenterar en litteraturstudie om indisk matematik. Den ger en kort översyn om ma- tematikens historia i Indien under många hundra år och handlar om de olika indiska metoderna för att lösa olika typer av ekvationer. Uppsatsen kommer att delas in i fyra avsnitt: 1) Kvadratisk och kubisk extraktion av Aryabhata 2) Kuttaka av Aryabhata för att lösa den linjära ekvationen på formen 푐 = 푎푥 + 푏푦 3) Bhavana-metoden av Brahmagupta för att lösa kvadratisk ekvation på formen 퐷푥2 + 1 = 푦2 4) Chakravala-metoden som är en annan metod av Bhaskara och Jayadeva för att lösa kvadratisk ekvation 퐷푥2 + 1 = 푦2.
    [Show full text]
  • International Conference on the 900Th Birth Anniversary of Bhāskarācārya
    International Conference on the 900th Birth Anniversary of Bhāskarācārya 19, 20, 21 September 2014 ABSTRACTS Vidya Prasarak Mandai Vishnu Nagar, Naupada, Thane 400602 2 I BHĀSKARĀCĀRYA’S LIFE AND TIMES A. P. JAMKHEDKAR, Mumbai. ‘Learning and Patronage in 12th -13th Century A.D.: Bhaskarācarya and the Śāndilya Family, A Case Study’ … … … … … 5 II BHASKARĀCĀRYA’S POETIC GENIUS Pierre-Sylvain FILLIOZAT, Paris. ‘The poetical face of the mathematical and astronomical works of Bhāskarācārya’ … … … … … … … … 7 K. S. BALASUBRAMANIAN, Chennai. ‘Bhāskarācārya as a Poet’ … … … … 8 GEETHAKUMARI, K. K., Calicut. ‘Contribution of Līlāvatī to Prosody’ … … … 9 Poonam GHAI, Dhampur. ‘Līlāvatī men Kāvya-saundarya’ … … … … … … 10 III THE LĪLĀVATĪ K. RAMASUBRAMANIAN, Mumbai. ‘The Līlā of the Līlāvatī: A Beautiful Blend of Arithmetic, Geometry and Poetry’ … … … … … … … … … 11 PADMAVATAMMA, Mysore. ‘A Comparative Study of Bhāskarācārya’s Līlāvatī and Mahāvīrācārya’s Gaṇitasārasaṁgraha’… … … … … … … … 11 K. RAMAKALYANI, Chennai. ‘Gaṇeśa Daivajña’s Upapattis on Līlāvatī’ … … 12 Anil NARAYANAN, N., Calicut. ‘Parameswara’s Unpublished Commentary on Lilavati: Perceptions and Problems’ … … … … … … … … … … … 13 P. RAJASEKHAR & LAKSHMI, V. Calicut. ‘Līlāvatī and Kerala School of Mathematics’ 14 N. K. Sundareswaran & P. M. Vrinda, Calicut. ‘Malayalam Commentaries on Līlāvatī – A Survey of Manuscripts in Kerala’ … … … … … … 15 Shrikrishna G. DANI, Mumbai. ‘Mensuration of Quadrilaterals in Lilavati’ … … 16 Takanori KUSUBA, Osaka. ‘Aṅkapāśa in the Līlāvatī’ … … … … … … … 17 Medha Srikanth LIMAYE, Mumbai. Use of Bhūta-Saṅkhyās (Object Numerals) in Līlāvatī of Bhāskarācārya’ … … … … … … … … … … 17 Sreeramula Rajeswara SARMA, Düsseldorf. ‘The Legend of Līlāvatī’ … … … 17 IV THE BĪJAGAṆITA Sita Sundar RAM, Chennai. ‘Bījagaṇita of Bhāskara’ … … … … … … … 19 K. VIDYUTA, Chennai. ‘Sūryaprakāśa of Sūryadāsa – a Review’ … … … … 20 Veena SHINDE-DEORE & Manisha M. ACHARYA, Mumbai. ‘Bhaskaracharya and Varga Prakriti: the equations of the type ax2 + b = cy2’ … … … … … 20 K.
    [Show full text]
  • Diophantine Analysis
    The Second International Conference “Problems of Cybernetics and Informatics” September 10-12, 2008, Baku, Azerbaijan. The plenary paper www.pci2008.science.az/5/01.pdf DIOPHANTINE ANALYSIS Djangir Babayev Cox Associates, Denver, CO, USA 4834 Macintosh Place, Boulder CO, USA, 80301 [email protected] Diophantus was a Greek Mathematician sometimes known as”the father of algebra”, who lived in 3rd century of AD (200-284). He is best known for his Arithmetica. Diophantus studied solving equations or systems of equations with numbers of equations less than number of variables for finding integer solutions and was first to try to develop algebraic notations. His works had enormous influence on the development of Number Theory. Modern Diophantine Analysis embraces mathematics of solving equations and problems in integer numbers. While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, but they can be extremely difficult to attack, and many open problems and conjectures remain. Division algorithm. If a>b>0, integers, then a=bq+r, where q and r are integers and b>r ≥ 0. Let gcd(a,b) be the greatest common divisor of integers a and b. Theorem 1. gcd(a,b) = gcd(b,r) P R O O F. Suppose gcd(a,b) =d. Then a=sd and b=td. r=a-bq=sd-bqd=(s-bq)d. This implies gcd(b,r)=d and completes the proof. ▓ Euclids algorithm for determining gcd(a,b) works based on gcd(a,b) = gcd(b,r) , and b=r q2 +r2 , r> r2 ≥ 0 by repeatedly applying the division algorithm to the last two remainders.
    [Show full text]
  • Euler and Lagrange on Pell's Equation
    Euler and Lagrange on Pell's Equation Joshua Aaron McGill ID: 260352168 December 3, 2010 Introduction The field of number theory is notorious for yielding immensely difficult problems that are deceptively easy to state. One such problem, known as Pell's Equation, was studied by some of the greatest mathematicians in history and was not fully solved until the 18th century.1 Here we will show early attempts at solving this equation by the Indian Mathematician Brahmagupta (598 − 668 CE) who forged an identity still used in modern mathematics. Bhaskara II (1114 − 1185 CE), used this identity to develope the Chakravala method which he then used as an efficient means of computing solutions to Pell's Equation. Nearly 600 years later, Leonard Euler (1707 − 1783 CE) gave an in depth analysis of Pell's Equation and other Diophantine equations in his book Elements of Algebra (1765). This method leaned on earlier attempts by Wallis and is in some ways reminiscent of the cyclic Chakravala method presented by Bhaskara II. The difference lies in the fact that Euler was far more rigorous in his description and used the principle of induction and a precursor to ”infinite descent," whereas Bhaskara II simply stated his results-we are never told how he discovered his ingenious method. Six years after Elements of Algebra was published, a thirty-five-year-old Joseph Lagrange (1736 − 1813 CE) made significant additions to this book in a volume entitled Additions. As well as providing some corrections and comments, Lagrange includes a proof of the reducibility of other quadratic diophantine equations to the Pell Equation.
    [Show full text]
  • EXPLORING the HISTORY of INDIAN MATHEMATICS December 4-6, 2017 IIT GANDHINAGAR
    International Conference on EXPLORING THE HISTORY OF INDIAN MATHEMATICS December 4-6, 2017 IIT GANDHINAGAR International Conference on Exploring the History of Indian Mathematics List of Speakers P.P. DIVAKARAN; IUCAA, Pune & CMI, Chennai, INDIA KIM PLOFKER; Union College, Schenectady, NY 12308, USA AMARTYA KUMAR DUTTA, Indian Statistical Institute, Kolkata, INDIA M.S. SRIRAM; Prof. K.V. Sarma Research Foundation, Adyar, Chennai, INDIA PARTHASARATHI MUKHOPADHYAY; Department of Mathematics Ramakrishna Mission Residential College (Autn.), Narendrapur, Kolkata, INDIA K. RAMASUBRAMANIAN; IIT Bombay, INDIA M.D.SRINIVAS; Centre for Policy Studies, Chennai, INDIA VENKETESWARA PAI R; Department of Humanities and Social Sciences, Indian Institute of Science Education and Research (IISER), Pune, INDIA S.G. DANI; UM-DAE Centre for Excellence in Basic Sciences (CBS), Mumbai 400032, INDIA CLEMENCY MONTELLE; University of Canterbury, NEW ZEALAND ATUL DIXIT; IIT Gandhinagar, INDIA MICHEL DANINO; IIT Gandhinagar, INDIA ANIL NARAYANAN, N.; Chinmaya Sanskrit University, Kerala,INDIA ADITYA KOLACHANA; IIT Bombay ARADHANA ANAND; Delhi Public School, Bangalore South, INDIA K. MAHESH; IIT Bombay R. GONDHALEKAR; VPM’s Academy of International Education and Research, Thane, INDIA JAMBUGAHAPITIYE DHAMMALOKA; University of Canterbury, NEW ZEALAND & Department of Classical Languages, University of Peradeniya, Sri Lanka. PRIYAMVADA, N.; Guruvayurappan Temple in Morganville, New Jersey, USA SIVANANDAN D.S.; Centre for Spiritual Studies, Amritapuri Campus, Amrita Uni- versity, INDIA International Conference on Exploring the History of Indian Mathematics Title and abstract Invited Talk 1. Title: Two masters: Aryabhata and Madhava, a reassessment Speaker: P.P. DIVAKARAN; IUCAA, Pune & CMI, Chennai, INDIA Abstract: There are, still, fresh insights to be gained from a study of these two masters, relating both to their mathematics – the sources of their creative ideas and the legacies they left behind – and their personal histories.
    [Show full text]
  • Number Theory and Its New Developments
    International Journal of Research in Engineering, Science and Management 327 Volume-2, Issue-7, July-2019 www.ijresm.com | ISSN (Online): 2581-5792 Number Theory and its New Developments Mohammod Noor1, Venkata Srinivasa Rao Naralasetty2 1,2Lecturer, Department of Mathematics, AG & SG Siddhratha Degree College, Vuyyuru, India Abstract: This paper presents an overview on number theory used, the triples were first constructed and then reordered by , and its new developments. presumably for actual use as a "table", for example, with a view to applications. It is not known what these applications may Keywords: number theory have been, or whether there could have been any; Babylonian astronomy, for example, truly came into its own only later. It 1. Introduction has been suggested instead that the table was a source of Number theory (or arithmetic or higher arithmetic in older numerical examples for school problems. While Babylonian usage) is a branch of pure mathematics devoted primarily to the number theory—or what survives of Babylonian study of the integers. German mathematician Carl Friedrich mathematics that can be called thus—consists of this single, Gauss (1777–1855) said, "Mathematics is the queen of the striking fragment, Babylonian algebra (in the secondary-school sciences-and number theory is the queen of mathematics." sense of "algebra") was exceptionally well developed. Late Number theorists study prime numbers as well as the properties Neoplatonic sources state that Pythagoras learned mathematics of objects made out of integers (for example, rational numbers) from the Babylonians. Much earlier sources state or defined as generalizations of the integers (for that Thales and Pythagoras traveled and studied in Egypt.
    [Show full text]
  • Session II – Unit I – Fundamentals
    Tutor:- Prof. Avadhoot S. Joshi Subject:- Design & Analysis of Algorithms Session II Unit I - Fundamentals • Algorithm Design – II • Algorithm as a Technology • What kinds of problems are solved by algorithms? • Evolution of Algorithms • Timeline of Algorithms Algorithm Design – II Algorithm Design (Cntd.) Figure briefly shows a sequence of steps one typically goes through in designing and analyzing an algorithm. Figure: Algorithm design and analysis process. Algorithm Design (Cntd.) Understand the problem: Before designing an algorithm is to understand completely the problem given. Read the problem’s description carefully and ask questions if you have any doubts about the problem, do a few small examples by hand, think about special cases, and ask questions again if needed. There are a few types of problems that arise in computing applications quite often. If the problem in question is one of them, you might be able to use a known algorithm for solving it. If you have to choose among several available algorithms, it helps to understand how such an algorithm works and to know its strengths and weaknesses. But often you will not find a readily available algorithm and will have to design your own. An input to an algorithm specifies an instance of the problem the algorithm solves. It is very important to specify exactly the set of instances the algorithm needs to handle. Algorithm Design (Cntd.) Ascertaining the Capabilities of the Computational Device: After understanding the problem you need to ascertain the capabilities of the computational device the algorithm is intended for. The vast majority of algorithms in use today are still destined to be programmed for a computer closely resembling the von Neumann machine—a computer architecture outlined by the prominent Hungarian-American mathematician John von Neumann (1903–1957), in collaboration with A.
    [Show full text]
  • Number Theory in Geometry
    Number Theory in Geometry Gopal Prasad Myhill Lectures, October 18-19-20, 2016, at SUNY, Buffalo It is a great honor to deliver Myhill lectures. I thank Mohan Ramachandran, David Hemmer and other colleagues here for the invitation to deliver these lectures this year. I had spent a month in spring of 1973 here to attend courses on Algebraic Geometry and Algebraic Groups given by Alexander Grothendieck. I have very fond memory of that stay and of my interaction with Grothendieck. The topic of my talk is number theory in geometry. I will say a few words about history of mathematics to show how geometry has played an important role in the development of mathematics. I will also mention some contributions of Indian Mathematicians of antiquity since these are not widely known. Humans, and apparently some animals too, can and need to count. We count discrete objects, such as siblings, animals, friends, trees, . Therefore, integers have existed in our consciousness for a very very long time. Babylonians had discovered the place-value representation of integers over 3 millennia BCE; they used base 601, so they needed 60 distinct symbols to write the numbers. Historians of mathematics believe that Indians independently came up with place-value representation to base 10 during the Vedic period (c. 1500 BCE). It is not clear if Indus-valley civilization had numerals in their writing; some experts have opined that they indeed had, and that they used 8 as the base. In any case, base 10 is much simpler than base 60 { imagine a child memorizing the multiplication table up to 59×59.
    [Show full text]
  • Number Theory and Combinatorics
    Number Theory and Combinatorics B SURY Stat-Math Unit, Indian Statistical Institute, Bengaluru, India. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the publisher. c Indian Academy of Sciences 2017 Reproduced from Resonance{journal of science education Reformatted by TNQ Books and Journals Pvt Ltd, www.tnq.co.in Published by Indian Academy of Sciences Foreword The Masterclass series of eBooks brings together pedagogical articles on single broad topics taken from Resonance, the Journal of Science Educa- tion, that has been published monthly by the Indian Academy of Sciences since January 1996. Primarily directed at students and teachers at the un- dergraduate level, the journal has brought out a wide spectrum of articles in a range of scientific disciplines. Articles in the journal are written in a style that makes them accessible to readers from diverse backgrounds, and in addition, they provide a useful source of instruction that is not always available in textbooks. The third book in the series, `Number Theory and Combinatorics', is by Prof. B Sury. A celebrated mathematician, Prof. Sury's career has largely been at the Tata Institute of Fundamental Research, Mumbai', and the Indian Statistical Institute, Bengaluru, where he is presently professor. He has contributed pedagogical articles regularly to Resonance, and his arti- cles on Number Theory and Combinatorics comprise the present book. He has also served for many years on the editorial board of Resonance. Prof.
    [Show full text]