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1729 (number)

From Wikipedia, the free encyclopedia
← 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

1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number. 1729 is also known as the Ramanujan number or Hardy–Ramanujan number.

In mathematics

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1729 is composite, its factors are 1, 7, 13, 19, 91, 133, 247, and 1729.[1] It is the multiplication of its first three smallest prime numbers .[2] Relatedly, it is the third Carmichael number,[3] and specifically 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 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based.[8] This is an example of a galactic algorithm.[9]

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.[10]

As a Ramanujan number

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

Ramujan numbers are named[11][12] after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. 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.[13] 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 .[12]

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.[14] A commemorative plaque now appears at the site of the Ramanujan–Hardy incident, at 2 Colinette Road in Putney.[15]

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.[16][17]

Notes

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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. ^ Anjema, Henry (1767). Table of divisors of all the natural numbers from 1. to 10000. p. 47. ISBN 9781140919421 – via the Internet Archive.
  2. ^ Sierpinski, W. (1998). Schinzel, A. (ed.). Elementary Theory of Numbers: Second English Edition. North-Holland. p. 233.
  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.
  5. ^ Chernick, J. (1939). "On Fermat's simple theorem" (PDF). Bulletin 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 (2nd ed.). Springer. p. 409. doi:10.1007/978-1-4419-8702-0. ISBN 978-1-4419-8702-0.
  8. ^ Harvey, David. "We've found a quicker way to multiply really big numbers". phys.org. Retrieved 2021-11-01.
  9. ^ Harvey, David; Hoeven, Joris van der (March 2019). "Integer multiplication in time ". HAL. hal-02070778.
  10. ^ Guy, Richard K. (2004). Unsolved Problems in Number Theory. Problem Books in Mathematics, Volume 1 (3rd ed.). Springer. doi:10.1007/978-0-387-26677-0. ISBN 0-387-20860-7.
    ISBN 978-0-387-26677-0 (eBook)
  11. ^ Edward, Graham; Ward, Thomas (2005). An Introduction to Number Theory. Springer. p. 117. ISBN 978-1-85233-917-3.
  12. ^ a b Lozano-Robledo, Álvaro (2019). Number Theory and Geometry: An Introduction to Arithmetic Geometry. American Mathematical Society. p. 413.
  13. ^ 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."
  14. ^ Kahle, Reinhard (2018). "Structure and Structures". In Piazza, Mario; Pulcini, Gabriele (eds.). Truth, Existence and Explanation: FilMat 2016 Studies in the Philosophy of Mathematics. p. 115. doi:10.1007/978-3-319-93342-9. ISBN 978-3-319-93342-9.
  15. ^ Marshall, Michael (24 February 2017). "A black plaque for Ramanujan, Hardy and 1,729". Good Thinking. Retrieved 7 March 2019.
  16. ^ 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.
  17. ^ 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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