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1729 (number)

(Redirected from Hardy–Ramanujan number)

1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It is also known as the Ramanujan number or Hardy–Ramanujan number, named after G. H. Hardy and Srinivasa Ramanujan.

← 1728 1729 1730 →
Cardinalone thousand seven hundred twenty-nine
Ordinal1729th
(one thousand seven hundred twenty-ninth)
Factorization7 × 13 × 19
Divisors1, 7, 13, 19, 91, 133, 247, 1729
Greek numeral,ΑΨΚΘ´
Roman numeralMDCCXXIX
Binary110110000012
Ternary21010013
Senary120016
Octal33018
Duodecimal100112
Hexadecimal6C116

As a natural number

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1729 is composite, the squarefree product of three prime numbers 7 × 13 × 19.[1] It has as factors 1, 7, 13, 19, 91, 133, 247, and 1729.[2] It is the third Carmichael number,[3] and the first Chernick–Carmichael number.[a] Furthermore, it is the first in the family of absolute Euler pseudoprimes, a subset of Carmichael numbers.[7]

1729 can be defined by summing each of its digits, multiplying by the resulting number with its digit permutably switched, a harshad number.[8] This property can be found in other number systems, such as the octal and hexadecimal. However, this does not work on binary number.[9] It is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based.[10] This is an example of a galactic algorithm.[11]

1729 can be expressed as the quadratic form. Investigating pairs of its distinct integer-valued that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729.[12]

Visually, 1729 can be found in other figurate numbers. It is the tenth centered cube number (a number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points), the nineteenth dodecagonal number (a figurate number in which the arrangement of points resembles the shape of a dodecagon), the thirteenth 24-gonal and the seventh 84-gonal number.[9][13]

As a Ramanujan number

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1729 can be expressed as a sum of two positive cubes in two ways, illustrated geometrically.

1729 is also known as Ramanujan number or Hardy–Ramanujan number, named after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital.[14][15] In their conversation, Hardy stated that the number 1729 from a taxicab he rode was a "dull" number and "hopefully it is not unfavourable omen", but Ramanujan otherwise stated it is a number that can be expressed as the sum of two cubic numbers in two different ways.[16] This conversation in the aftermath led to a new class of numbers known as the taxicab number. 1729 is the second taxicab number, expressed as   and  .[15]

1729 was also found in one of Ramanujan's notebooks dated years before the incident and was noted by French mathematician Frénicle de Bessy in 1657.[17] A commemorative plaque now appears at the site of the Ramanujan–Hardy incident, at 2 Colinette Road in Putney.[18]

The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined, in reference to Fermat's Last Theorem, as numbers of the form  , which are also expressible as the sum of two other cubes.[19][20]

See also

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Explanatory footnotes

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  1. ^ It is a number in which Chernick (1939) expressed Carmichael number as the product of three prime numbers  .[4][5][6]

References

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  1. ^ Sierpinski, W. (1998). Schinzel, A. (ed.). Elementary Theory of Numbers: Second English Edition. North-Holland. p. 233. ISBN 978-0-08-096019-7.
  2. ^ Anjema, Henry (1767). Table of divisors of all the natural numbers from 1. to 10000. p. 47. ISBN 9781140919421 – via the Internet Archive.
  3. ^ Koshy, Thomas (2007). Elementary Number Theory with Applications (2nd ed.). Academic Press. p. 340. ISBN 978-0-12-372487-8.
  4. ^ Deza, Elena (2022). Mersenne Numbers And Fermat Numbers. World Scientific. p. 51. ISBN 978-981-12-3033-2.
  5. ^ Chernick, J. (1939). "On Fermat's simple theorem" (PDF). Bulletin of the American Mathematical Society. 45 (4): 269–274. doi:10.1090/S0002-9904-1939-06953-X.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A033502 (Carmichael number of the form  , where  ,  , and   are prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^ Childs, Lindsay N. (1995). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (2nd ed.). Springer. p. 409. doi:10.1007/978-1-4419-8702-0. ISBN 978-1-4419-8702-0.
  8. ^ Deza, Elena (2023). Perfect And Amicable Numbers. World Scientific. p. 411. ISBN 978-981-12-5964-7.
  9. ^ a b Deza, Michel-marie; Deza, Elena (2012). Figurate Numbers. World Scientific. p. 436. ISBN 978-981-4458-53-5.
  10. ^ Harvey, David. "We've found a quicker way to multiply really big numbers". phys.org. Retrieved 2021-11-01.
  11. ^ Harvey, David; Hoeven, Joris van der (March 2019). "Integer multiplication in time  ". HAL. hal-02070778.
  12. ^ Guy, Richard K. (2004). Unsolved Problems in Number Theory. Problem Books in Mathematics, Volume 1. Vol. 1 (3rd ed.). Springer. doi:10.1007/978-0-387-26677-0. ISBN 0-387-20860-7.
    ISBN 978-0-387-26677-0 (eBook)
  13. ^ Other sources on its figurate numbers can be found in the following:
  14. ^ Edward, Graham; Ward, Thomas (2005). An Introduction to Number Theory. Springer. p. 117. ISBN 978-1-85233-917-3.
  15. ^ a b Lozano-Robledo, Álvaro (2019). Number Theory and Geometry: An Introduction to Arithmetic Geometry. American Mathematical Society. p. 413. ISBN 978-1-4704-5016-8.
  16. ^ Hardy, G. H. (1940). Ramanujan. New York: Cambridge University Press. p. 12. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab No. 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."
  17. ^ Kahle, Reinhard (2018). "Structure and Structures". In Piazza, Mario; Pulcini, Gabriele (eds.). Truth, Existence and Explanation: FilMat 2016 Studies in the Philosophy of Mathematics. Boston Studies in the Philosophy and History of Science. Vol. 334. p. 115. doi:10.1007/978-3-319-93342-9. ISBN 978-3-319-93342-9.
  18. ^ Marshall, Michael (24 February 2017). "A black plaque for Ramanujan, Hardy and 1,729". Good Thinking. Retrieved 7 March 2019.
  19. ^ Ono, Ken; Aczel, Amir D. (2016). My Search for Ramanujan: How I Learned to Count. p. 228. doi:10.1007/978-3-319-25568-2. ISBN 978-3-319-25568-2.
  20. ^ Sloane, N. J. A. (ed.). "Sequence A050794 (Consider the Diophantine equation   ( ) or 'Fermat near misses')". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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