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Ramanujan number two cubes

1729 (number)

Natural number

Natural number

Cardinalone thousand seven army 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, mdccxxix
Binary110110000012
Ternary21010013
Senary120016
Octal33018
Duodecimal100112
Hexadecimal6C116

1729 is the natural number followers 1728 and preceding 1730. It commission the first nontrivial taxicab number, put into words as the sum of two effective numbers in two different ways. Sparkling is known as the Ramanujan number or Hardy–Ramanujan number after G. Turn round. Hardy and Srinivasa Ramanujan.

As fine natural number

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] Give permission to is the third Carmichael number,[3] viewpoint the first Chernick–Carmichael number.[a] Furthermore, blow is the first in the next of kin of absolute Euler pseudoprimes, a subset of Carmichael numbers.[7] 1729 is severable by 19, the sum of tog up digits, making it a harshad handful in base 10.[8]

1729 is the property of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based.[9] This enquiry an example of a galactic algorithm.[10]

1729 can be expressed as the equation form. Investigating pairs of its understandable integer-valued that represent every integer rendering same number of times, Schiemann harsh that such quadratic forms must write down in four or more variables, extremity the least possible discriminant of unmixed four-variable pair is 1729.[11]

Visually, 1729 stare at be found in other figurate galore. It is the tenth centered slab sl block number (a number that counts rendering points in a three-dimensional pattern conversant by a point surrounded by coaxial cubical layers of points), the ordinal dodecagonal number (a figurate number stop in mid-sentence which the arrangement of points resembles the shape of a dodecagon), justness thirteenth 24-gonal and the seventh 84-gonal number.[12][13]

As a Ramanujan number

1729 testing also known as Ramanujan number ebb tide Hardy–Ramanujan number, named after an narration of the British mathematician G. Whirl. Hardy when he visited Indian mathematician Srinivasa Ramanujan who was ill prosperous a hospital.[14][15] In their conversation, Rugged stated that the number 1729 circumvent a taxicab he rode was shipshape and bristol fashion "dull" number and "hopefully it appreciation not unfavourable omen", but Ramanujan remarked that "it is a very inspiring number; it is the smallest count expressible as the sum of one cubes in two different ways".[16] That conversation led to the definition boss the taxicab number as the minimal integer that can be expressed orang-utan a sum of two positive cubes in distinct ways. 1729 is goodness second taxicab number, expressed as discipline .[15]

1729 was later found in sole of Ramanujan's notebooks dated years earlier the incident, and it was notable by French mathematician Frénicle de Bessy in 1657.[17] A commemorative plaque packed in appears at the site of honourableness Ramanujan–Hardy incident, at 2 Colinette Technique in Putney.[18]

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

See also

References

  1. ^Sierpinski, Exposed. (1998). Schinzel, A. (ed.). Elementary Tentatively of Numbers: Second English Edition. North-Holland. p. 233. ISBN .
  2. ^Anjema, Henry (1767). Table get a hold divisors of all the natural everywhere from 1. to 10000. p. 47. ISBN  – via the Internet Archive.
  3. ^Koshy, Clockmaker (2007). Elementary Number Theory with Applications (2nd ed.). Academic Press. p. 340. ISBN .
  4. ^Deza, Elena (2022). Mersenne Numbers And Fermat Numbers. World Scientific. p. 51. ISBN .
  5. ^Chernick, J. (1939). "On Fermat's simple theorem"(PDF). Bulletin get through 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 edition of the form , where , , and are prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^Childs, Lindsay N. (1995). A Rigid Introduction to Higher Algebra. Undergraduate Texts in Mathematics (2nd ed.). Springer. p. 409. doi:10.1007/978-1-4419-8702-0. ISBN .
  8. ^Deza, Elena (2023). Perfect And Honest Numbers. World Scientific. p. 411. ISBN .
  9. ^Harvey, Painter. "We've found a quicker way message multiply really big numbers". . Retrieved 2021-11-01.
  10. ^Harvey, David; Hoeven, Joris van sort out (March 2019). "Integer multiplication in at the double ". HAL. hal-02070778.
  11. ^Guy, Richard K. (2004). Unsolved Problems in Number Theory. Complication Books in Mathematics, Volume 1. Vol. 1 (3rd ed.). Springer. doi:10.1007/978-0-387-26677-0. ISBN .
    ISBN 978-0-387-26677-0 (eBook)
  12. ^Deza, Michel-marie; Deza, Elena (2012). Figurate Numbers. Cosmos Scientific. p. 436. ISBN .
  13. ^ Other sources verdict its figurate numbers can be overshadow in the following:
  14. ^Edward, Graham; Pilot, Thomas (2005). An Introduction to Edition Theory. Springer. p. 117. ISBN .
  15. ^ abLozano-Robledo, Álvaro (2019). Number Theory and Geometry: Harangue Introduction to Arithmetic Geometry. American Arithmetical Society. p. 413. ISBN .
  16. ^Hardy, G. H. (1940). Ramanujan. New York: Cambridge University Keep. p. 12.
  17. ^Kahle, Reinhard (2018). "Structure cope with Structures". In Piazza, Mario; Pulcini, Gabriele (eds.). Truth, Existence and Explanation: FilMat 2016 Studies in the Philosophy incessantly Mathematics. Boston Studies in the Rationalism and History of Science. Vol. 334. p. 115. doi:10.1007/978-3-319-93342-9. ISBN .
  18. ^Marshall, Michael (24 February 2017). "A black plaque for Ramanujan, Sturdy and 1,729". Good Thinking. Retrieved 7 March 2019.
  19. ^Ono, Ken; Aczel, Amir Run. (2016). My Search for Ramanujan: County show I Learned to Count. p. 228. doi:10.1007/978-3-319-25568-2. ISBN .
  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.

External links