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Ramanujan

	Taxicab number - Wikipedia, the free encyclopedia
		In mathematics, the n-th taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number which can be expressed as a sum of two positive cubes in n distinct ways, up to order of summands.
		I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen.
		Hardy and E. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford University Press, London and NY, 1954, Thm.
	en.wikipedia.org /wiki/Taxicab_number (410 words)

	1729 (number) - Wikipedia, the free encyclopedia
		I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.
		It is a centered cube number, as well as a dodecagonal number, a 24-gonal and 84-gonal number.
		Because in base 10 the number 1729 is divisible by the sum of its digits, it is a Harshad number.
	en.wikipedia.org /wiki/1729_(number) (690 words)

	PlanetMath: number theory
		Number theory is the branch of math concerned with the study of the integers, and of the objects and structures that naturally arise from their study.
		Algebraic number theory can either be defined as the study of algebraic numbers or as an algebraic study of number theory (depending on how you associate in English).
		Finally, computational number theory is the study of computations with numbers, developing algorithms to calculate things such as factorizations, discrete logarithms, numbers of points on curves, class groups and cohomology groups.
	planetmath.org /encyclopedia/NumberTheory.html (696 words)

	SRINIVASA RAMANUJAN
		Ramanujan was named a research scholar at the University of Madras, receiving double his clerk's salary and required only to submit quarterly reports on his work.
		Ramanujan's arrival at Cambridge was the beginning of a very successful five-year collaboration with Hardy.
		Ramanujan had always lived in a tropical climate and had his mother (later his wife) to cook for him: now he faced the English winter, and he had to do all his own cooking to adhere to his caste's strict dietary rules.
	www.usna.edu /Users/math/meh/ramanujan.html (921 words)

	The Dullness of 1729 (Site not responding. Last check: 2007-10-21)
		The famous anecdote is that during one visit to Ramanujan in the hospital at Putney, Hardy mentioned that the number of the taxi cab that had brought him was 1729, which, as numbers go, Hardy thought was "rather a dull one".
		At this, Ramanujan perked up, and said "No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways." This was the sort of thing that prompted Littlewood to say "every positive integer was one of [Ramanujans'] personal friends".
		In a chapter entitled "Lucky Numbers" he tells of going into a small restaurant in Brazil to eat lunch, and he's the only customer in the place so he has four waiters standing around him, and then a Japanese man enters the restaurant, and he is selling abacuses.
	www.mathpages.com /home/kmath028.htm (853 words)

	PI - Ramanujan's Method (Site not responding. Last check: 2007-10-21)
		Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function on his own.
		Srinivasa Ramanujan was elected to the Royal Society of London in 1918.
		Ramanujan's health deteriorated rapidly in England, due perhaps to the unfamiliar climate, food, and to the isolation he felt in a culture which was largely foreign to him.
	ic.net /~jnbohr/java/Ramanujan.html (253 words)

	Hardy
		Hardy was elected a fellow of Trinity in 1900 then, in 1901, he was awarded a Smith's prize jointly with J H Jeans 'with unspecified relative merit'.
		A major change in Hardy's work came about in 1911 when he began his collaboration with J E Littlewood which was to last 35 years.
		It was a collaboration in which Hardy acknowledged Littlewood's greater technical mathematical skills, but at the same time Hardy brought great talents of mathematical insight and a great ability to write their work up in papers with great clarity.
	www-history.mcs.st-and.ac.uk /history/Mathematicians/Hardy.html (2430 words)

	Ramanujan (Site not responding. Last check: 2007-10-21)
		Ramanujan was awarded in 1916 the B.A. Degree by research of the Cambridge University.
		The Ramanujan Institute for Advanced Study in Mathematics of the University of Madras is situated at a short distance from the famed Marina Beach and is close to the Administrative Buildings of the University and its Library.
		A bust of Ramanujan, sculpted by Paul Granlund was presented to her and it is now with her adopted son Mr.
	www.math.buffalo.edu /~aagarwal/RAMANUJAN/ramanujan.html (1178 words)

	Ramanujan (Site not responding. Last check: 2007-10-21)
		Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904.
		Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education.
		Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Ramanujan.html (2639 words)

	Rediscovering Ramanujan
		This is a dull number." Ramanujan replied: "No, it is a very interestin g number; it is the smallest number expressible as a sum of two cubes in two different ways." Berndt believes that this was no flash of insight, as is commonly thought.
		Hardy was a number theorist but he was also into analysis.
		Ramanujan had a number of conjectures in regard to this formula and one is still unproven.
	www.flonnet.com /fl1617/16170810.htm (3014 words)

	Dream 2047 December issue (Site not responding. Last check: 2007-10-21)
		Ramanujan was awarded the B.A. degree in March 1916 for his work on ‘Highly composite Numbers’ which was published as a paper in the Journal of the London Mathematical Society.
		Ramanujan was a mathematical genius in his own right on the basis of his work alone.
		Of course, as Hardy observed Ramanujan “combined a power of generalization, a feeling for form and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his peculiar field, without a rival in his day.
	www.vigyanprasar.com /dream/dec2001/article1.htm (3073 words)

	Ivars Peterson's MathTrek - Taxicab Numbers
		His friend G.H. Hardy (1877-1947) once remarked that the taxi by which he had arrived had a "dull" number--1729, or 7 x 13 x 19.
		Ramanujan was quick to point out that 1729 is actually a "very interesting" number.
		The eighth cabtaxi number is now known, and the ninth must have at least 19 digits.
	www.maa.org /mathland/mathtrek_07_22_02.html (734 words)

	List of numbers - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		Other numbers that are notable for their mathematical properties or cultural meanings include:
		This is a table of English names for positive rational numbers less than 1.
		Keep in mind that rational numbers like 0.12 can be represented in infinite ways, e.g.
	www.eastcleveland.us /project/wikipedia/index.php/List_of_numbers (712 words)

	Ramanujan
		Ramanujan was one of India's greatest mathematical geniuses.
		Truly, the life of Ramanujan in the words of C.P. Snow: ``is an admirable story and one which showers credit on nearly everyone''.
		G.H. Hardy and these as well as his earlier publications before he set sail to England are all contained in the ``Collected Papers of Srinivasa Ramanujan'', referred earlier.
	www.imsc.res.in /~rao/ramanujan.html (1202 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Once, when Ramanujan was very ill, his colleague and friend G. Hardy took a taxi to the hospital to visit him.
		It is the smallest number expressible as a sum of two cubes in two different ways!” Finding numbers that are the sum of one pair of cubes is easy.
		As Ramanujan said, the smallest number which can be stated as the sum of two pairs of cubes is 1729.
	www.manchester.gov.uk /education/diversity/ema/blackhistory/the_curious_cab.doc (287 words)

	Ramanujan by Hardy (Site not responding. Last check: 2007-10-21)
		The difficulties in judging Ramanujan are clear --- he was an Indian, I am an Englishman, and the two parties have always found it hard to understand one another.
		Srinivasa Aiyangar Ramanujan was born in 1887 in a poor Brahmin family at Erode near Kumbakonam, a fair sized town in the Tanjore district of Tamil Nadu.
		Ramanujan went through the entire book methodically and excitedly, proving its theorems by himself, often as he got up in the morn.
	uzweb.uz.ac.zw /science/maths/zimaths/ramanhdy.htm (1224 words)

	INDOlink - NRI News - Ramanujan’s “Lost Notebook” Astounds Americans
		Alladi, during the 1987 Ramanujan Centennial, the printed form of Ramanujan's Lost Notebook by Springer-Narosa was released by Prime Minister Rajiv Gandhi, who presented the first copy to Janaki Ammal Ramanujan, the late widow of Srinivasa Ramanujan, and the second copy to Professor Andrews in recognition of his contributions.
		Bruce Berndt, an analytic number theorist with strong interests in several related areas of classical analysis, has devoted 31 years of his research to proving the claims left in three notebooks and a "lost notebook" by the Indian genius upon his death in 1920.
		The 600 formulae that Ramanujan jotted down on loose sheets of paper during the one year he was in India, after he returned from Cambridge, are the contents of the `Lost' Note Book found by Andrews in 1976.
	www.indolink.com /displayArticleS.php?id=021705075416 (1976 words)

	The CTK Exchange Forums
		Hardy remarked that he took note of the number of the taxi in which he had arrived, but unfortunately it appeared to be a rather mundane number: 1729.
		Ramanujan instantly replied, "On the contrary, 1729 is a most interesting number.
		However, fortune smiles on Feynman because the number that is chosen to calculate the cube root of is 1729.03.
	www.cut-the-knot.com /htdocs/dcforum/DCForumID4/601.shtml (444 words)

	Notable Properties of Specific Numbers at MROB (Site not responding. Last check: 2007-10-21)
		The number of white cows was (1/3+1/4) the total number of the fl herd.
		The number of yellow cows was (1/6+1/7) the total number of the white herd.
		If you can accurately tell, O stranger, the total number of cattle of the Sun, including the number of cows and bulls in each color, you would not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.
	home.earthlink.net /~mrob/pub/math/numbers-14.html (2178 words)

	The Fifth Taxicab Number is 48988659276962496
		The nth taxicab number is the least number which can be expressed as a sum of two positive cubes in n distinct ways, up to order of summands.
		A brief history of taxicab numbers is given, along with a description of the computer search used by the author to find the 5th taxicab number, 48988659276962496.
		I had ridden in taxi-cab No. 1729, and remarked that the number (7.13.19) seemed to be rather a dull one, and that I hoped it was not an unfavourable omen.
	www.cs.uwaterloo.ca /journals/JIS/wilson10.html (2186 words)

	Mathematical Curiosity Shop (Site not responding. Last check: 2007-10-21)
		It is the smallest number expressible as a sum of two cubes in two different ways'.
		Ramanujan was fascinated by numbers, and it is no surprise that he found some weird approximations for
		The number 25920 is not of course exact, but is probably accurate to the nearest ten years.
	www.mth.uct.ac.za /digest/curious.html (222 words)

	Taxicab Numbers (Site not responding. Last check: 2007-10-21)
		He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it.
		It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen.
		In memory of this incident, the least number which is the sum of two positive cubes in n different ways is called the nth taxicab number, which I will denote Taxicab(n).
	pi.lacim.uqam.ca /eng/problem_en.html (267 words)

	Hinduism Today \| Ramanuju\| October/November/December, 2003
		Stubborn and religious, Ramanujan (1887-1920) saw the divine in the dance of numbers; an acquaintance said of him that "every integer was his personal friend." During an illness in England, Hardy visited Ramanujan in the hospital.
		When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number is actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes!
		Ramanujan's work has been used to help unravel knots as varied as polymer chemistry and cancer, yet how he arrived at his theorems is still unknown, unless one takes him at his word--as most Hindus would--about the Goddess.
	www.hinduism-today.com /archives/2003/10-12/60-61_ramanuju_play.shtml (1126 words)

	MATHEWS: Taxicab Numbers
		Generalizing the above observation the nth taxicab number Ta(n) is defined as the least integer expressible in n different ways as the sum of two positive cubes.
		In the famous book An Introduction to the Theory of Numbers of Hardy and Wright it is proven that the nth taxicab number exists but the proof is of no use in finding the number.
		Hardy, G.H. and Wright, E.M.: An Introduction to the Theory of Numbers,
	www.wschnei.de /number-theory/taxicab-numbers.html (384 words)

	Amazon.ca: Books: An Introduction to the Theory of Numbers (Site not responding. Last check: 2007-10-21)
		Hardy gives actual intuitive motivation for almost all of the theorems in the book (intuition is often overlooked by mathematical authors who use the confusing traditional "theorem-proof" approach), and his proofs are elegant and easy to follow.
		While modern results in the area of algorithmic number theory are not presented nor is a systematic presentation of number theory given (it is not a textbook), it contains a flavor, inspiration and feel that is completely unique.
		Primarily intended as a textbook for a one semester number theory course.
	www.amazon.ca /exec/obidos/ASIN/0198531710 (982 words)

	Home Page
		His achievements include Hardy-Ramanujan-Littlewood circle method in number theory, Roger-Ramanujan's identities in partition of numbers, work on algebra of inequalities, elliptic functions, continued fractions, partial sums and products of hypergeometric series, etc. He was the second Indian to be elected Fellow of the Royal Society in February, 1918.
		When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes: 1729=13+123=93+103.
		Unfortunately, Ramanujan's health deteriorated due to tuberculosis, and he returnted to India in 1919.
	mathsindia.4t.com (544 words)

	No. 495: Ramanujan
		Ramanujan did it from his sickbed without blinking.
		Ramanujan was born to a poor family in South India in 1887.
		Hardy would've ignored the letter, but he took a moment to glance at 120 theorems Ramanujan had included.
	www.uh.edu /engines/epi495.htm (439 words)

	[No title]
		There are numbers called pseudo-primes or Carmichael numbers: these are composite numbers for which a^(N-1) mod N = 1 for all a in Z* sub N.
		ASIDE: 1729 = The Hardy-Ramanujan "Taxi" number is Carmichael number: = 71319 = least number expressible as sum of two positive integral cubes in two different ways: 1729 = 9^3+10^3 = 1^3+12^3 Fortunately, there is an efficient test for Carmichael numbers, so modulo this test, we have an efficient randomizing test for primality.
		Let pi(N) = number of primes less than N. Then pi(N) is asymptotic to N/(ln N), meaning that the ratio pi(N) lim --------- = 1 N -> infinity N/(ln N) Typical notation: pi(N) ~ N/ln(N).
	www.cs.cmu.edu /afs/cs.cmu.edu/academic/class/15451-f04/www/lectures/lect1118.txt (550 words)

	Edge: What Are Numbers, Really? - Stanislas Dehaene [page 4]
		The most crucial one is, of course, the issue of how mathematical education modifies this representation, and why some children develop a talent for arithmetic and mathematics while others (many of us!) remain innumerate.
		I think that the acquisition of a language for numbers is crucial, and it is at that stage that cultural and educational differences appear.
		The prodigious Indian mathematician Ramanujan was slowly dying of tuberculosis when his colleague Hardy came to visit him and, not knowing what to say, made the following reflection: "The taxi that I hired to come here bore the number 1729.
	www.edge.org /3rd_culture/dehaene/dehaene_p4.html (631 words)

	[No title]
		I am a member of the Department of Pure and Applied Mathematics of the University of Padova, where I am Assistant Professor (the Italian name of my profession is Ricercatore).
		My main research interest is in Analytic Number Theory, but I officially work in the field of Mathematical Analysis.
		Primes is in P! The website of one of the most important recent results in Computational Number Theory by M. Agrawal, N. Kayal, N. Saxena.
	www.math.unipd.it /~languasc (291 words)

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