Ramanujan number

A British taxi numbered 1729 sparked the most famous anecdote in math and led to the origin of "taxi-cab numbers." The incident is included in an upcoming biopic of Ramanujan, "The Man Who Knew Infinity," featuring Dev Patel in the lead role. Above is a still from the movie. (Pressman Films.) Taxi-cab numbers, among the most beloved integers in math, trace their origins to 1918 and what seemed like a casual insight by the Indian genius Srinivasa Ramanujan. Now mathematicians at Emory University have discovered that Ramanujan did not just identify the first taxi-cab number - 1729 - and its quirky properties. He showed how the number relates to elliptic curves and K3 surfaces - objects important today in string theory and quantum physics. "We've found that Ramanujan actually discovered a K3 surface more than 30 years before others started studying K3 surfaces and they were even named," says Ken Ono, a Ono and his graduate student Sarah Trebat-Leder are publishing a paper about these new insights in the journal Research in Number Theory. Their paper also demonstrates how one of Ramanujan's formulas associated with the taxi-cab number can reveal secrets of elliptic curves. "We were able to tie the record for finding certain elliptic curves with an unexpected number of points, or solutions, without doing any heavy lifting at all," Ono says. "Ramanujan's formula, which he wrote on his deathbed in 1919, is that ingenious. It's as though he left a magic key for the mathematicians ...

In nth taxicab number, typically denoted Ta( n) or Taxicab( n), also called the nth Ramanujan–Hardy number, is defined as the smallest integer that can be expressed as a sum of two positive n distinct ways. The most famous taxicab number is 3 + 12 3 = 9 3 + 10 3. The name is derived from a conversation in about 1919 involving I remember once going to see him [Ramanujan] when he was lying ill at History and definition [ ] The concept was first mentioned in 1657 by n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are the smallest possible and so it cannot be used to find the actual value of Ta( n). The taxicab numbers subsequent to 1729 were found with the help of computers. The restriction of the n distinct ways. The concept of a Known taxicab numbers [ ] So far, the following 6 taxicab numbers are known: Ta ⁡ ( 1 ) = 2 = 1 3 + 1 3 Cubefree taxicab numbers [ ] A more restrictive taxicab problem requires that the taxicab number be 3. When a cubefree taxicab number T is written as T = x 3+ y 3, the numbers x and y must be relatively prime. Among the taxicab numbers Ta( n) listed above, only Ta(1) and Ta(2) are cubefree taxicab numbers. The smallest cubefree taxicab number with three representations was discovered by 15170835645 = 517 3 + 2468 3 = 709 3 + 2456 3 = 1733 3 + 2152 3. The smallest cubefree taxicab number with four representations was discovered by Stu...

(3) which is associated with a story told about Ramanujan by G.H.Hardy (Hofstadter 1989, Kanigel 1991, Snow 1993). This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Futurama episode DVD 2ACV02 (Greenwald; left figure), as well as the robot character Bender's serial number, as portrayed in a Christmas card in the episode Xmas Story (Volume 2 DVD, Georgoulias et al. 2004; right figure). However, this property was also known as early as 1657 by F.de Bessy (Berndt and Bhargava 1993, Guy 1994). Leech (1957) found (25) (Calude et al. 2003, Hollerbach 2008). Hardy and Wright (Theorem 412, 1979) show that the number of such sums can be made arbitrarily large but, updating Guy (1994) with Wilson's result, the least example is not known for six or more equal sums. Sloane defines a slightly different type of taxicab numbers, namely numbers which are sums of two cubes in two or more ways, the first few of which are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ... (OEIS More things to try: • • • References Berndt, B.C. and Bhargava, S. "Ramanujan--For Lowbrows." Am. Math. Monthly 100, 645-656, 1993. Butler, B. "Ramanujan Numbers and the Taxicab Problem." Calude,C.S.; Calude, E.; and Dinneen, M.J. "What Is the Value of Taxicab(6)?" J. Uni. Comp. Sci. 9, 1196-1203, 2003. Georgoulias, T.; Greenwald, S.J.; and Wichterich, M. "Futurama : Mathematics in the Year 3000." Math Horizons, 12-...

On the smallest non-interesting number The interesting number paradox is a humorous "Interestingness" concerning numbers is not a formal concept in normal terms, but an innate notion of "interestingness" seems to run among some Paradoxical nature [ ] Attempting to classify all numbers this way leads to a interesting and uninteresting sets seems to fail. Since the definition of interesting is usually a subjective, intuitive notion, it should be understood as a semi-humorous application of The paradox is alleviated if "interesting" is instead defined objectively: for example, the smallest natural number that does not appear in an entry of the all However, as there are many significant results in mathematics that make use of History [ ] In 1945, One might conjecture that there is an interesting fact concerning each of the positive integers. Here is a "proof by induction" that such is the case. Certainly, 1, which is a factor of each positive integer, qualifies, as do 2, the smallest prime; 3, the smallest odd prime; 4, Bieberbach's number; etc. Suppose the set S of positive integers concerning each of which there is no interesting fact is not vacuous, and let k be the smallest member of S. But this is a most interesting fact concerning k! Hence S has no smallest member and therefore is vacuous. Is the proof valid? In See also [ ] • • Notes [ ] • ^ a b Gardner, Martin (January 1958). "A collection of tantalizing fallacies of mathematics". Mathematical games. Scientific America...

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More things to try: • • • References Ramanujan, S. Ramanujan, S. "A Proof of Bertrand's Postulate." J. Indian Math. Soc. 11, 181-182, 1919. Sloane, N.J.A. Sequence Referenced on Wolfram|Alpha Cite this as: MathWorld--A Wolfram Web Resource, created by Subject classifications • • • • • • • • • • • • • • • • • • • • Created, developed and nurtured by Eric Weisstein at Wolfram Research

Ramanujan's manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. The equation expressing the near counter examples to Fermat's last theorem appears further up: α 3 + β 3 = γ 3 + (-1) n. Image courtesy A box of manuscripts and three notebooks. That's all that's left of the work of Srinivasa Ramanujan, an Indian mathematician who lived his remarkable but short life around the beginning of the twentieth century. Yet, that small stash of mathematical legacy still yields surprises. Two mathematicians of Emory University, Ramanujan's story is as inspiring as it is tragic. Born in 1887 in a small village around 400 km from Madras (now Chennai), Ramanujan developed a passion for mathematics at a young age, but had to pursue it mostly alone and in poverty. Until, in 1913, he decided to write a letter to the famous Cambridge number theorist The taxi-cab number The romanticism rubbed off on the number 1729, which plays a central role in the Hardy-Ramanujan story. "I remember once going to see [Ramanujan] when he was ill at Putney," Hardy wrote later. "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 unfavourable omen. 'No', he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'" What Ramanujan meant is that The anecdote gained the number 1729 fame in mathematical circles, but until re...