1729 (number)

Last updated
← 1728 17291730 →
Cardinal one thousand seven hundred twenty-nine
Ordinal 1729th
(one thousand seven hundred twenty-ninth)
Factorization 7 × 13 × 19
Divisors 1, 7, 13, 19, 91, 133, 247, 1729
Greek numeral ,ΑΨΚΘ´
Roman numeral MDCCXXIX
Binary 110110000012
Ternary 21010013
Senary 120016
Octal 33018
Duodecimal 100112
Hexadecimal 6C116

1729 is the natural number following 1728 and preceding 1730. It is notably the first nontrivial taxicab number.

Contents

In mathematics

Taxicab number

1729 as the sum of two positive cubes. Cube-sum-1729.png
1729 as the sum of two positive cubes.

1729 is the smallest nontrivial taxicab number, [1] and is known as the Hardy–Ramanujan number, [2] after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation: [3] [4] [5] [6]

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."

The two different ways are:

1729 = 13 + 123 = 93 + 103

The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729; 19 × 91 = 1729).

91 = 63 + (−5)3 = 43 + 33

1729 was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657. A commemorative plaque now appears at the site of the Ramanujan-Hardy incident, at 2 Colinette Road in Putney. [7]

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 1 + z3 which are also expressible as the sum of two other cubes (sequence A050794 in the OEIS ).

Other properties

1729 is a sphenic number. It is the third Carmichael number, and more specifically the first Chernick–Carmichael number (sequence A033502 in the OEIS ). Furthermore, it is the first in the family of absolute Euler pseudoprimes, which are a subset of Carmichael numbers.

1729 is the third Zeisel number. [8] It is a centered cube number, [9] as well as a dodecagonal number, [10] a 24-gonal [11] and 84-gonal number.

Investigating pairs of distinct integer-valued quadratic forms 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]

1729 is the lowest number which can be represented by a Loeschian quadratic form in four different ways with a and b positive integers. The integer pairs are (25,23), (32,15), (37,8) and (40,3). [13]

1729 is also the smallest integer side of an equilateral triangle for which there are three sets of non-equivalent points at integer distances from their vertices: {211, 1541, 1560}, {195, 1544, 1591}, and {824, 915, 1591}. [14]

1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based. [15] This is an example of a galactic algorithm.

See also

Related Research Articles

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

64 (sixty-four) is the natural number following 63 and preceding 65.

91 (ninety-one) is the natural number following 90 and preceding 92.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

700 is the natural number following 699 and preceding 701.

2000 is a natural number following 1999 and preceding 2001.

The interesting number paradox is a humorous paradox which arises from the attempt to classify every natural number as either "interesting" or "uninteresting". The paradox states that every natural number is interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting natural numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction.

<span class="mw-page-title-main">Taxicab number</span> Smallest integer expressable as a sum of two positive integer cubes in n distinct ways

In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103, also known as the Hardy-Ramanujan number.

4000 is the natural number following 3999 and preceding 4001. It is a decagonal number.

A Zeisel number, named after Helmut Zeisel, is a square-free integer k with at least three prime factors which fall into the pattern

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

10,000,000 is the natural number following 9,999,999 and preceding 10,000,001.

1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.

189 is the natural number following 188 and preceding 190.

744 is the natural number following 743 and preceding 745.

20,000 is the natural number that comes after 19,999 and before 20,001.

30,000 is the natural number that comes after 29,999 and before 30,001.

40,000 is the natural number that comes after 39,999 and before 40,001. It is the square of 200.

50,000 is the natural number that comes after 49,999 and before 50,001.

60,000 is the natural number that comes after 59,999 and before 60,001. It is a round number. It is the value of (75025).

References

  1. ↑ Higgins, Peter (2008). Number Story: From Counting to Cryptography . New York: Copernicus. p.  13. ISBN   978-1-84800-000-1.
  2. ↑ "Hardy-Ramanujan Number". Wolfram Mathworld.
  3. ↑ Quotations by Hardy Archived 2012-07-16 at the Wayback Machine
  4. ↑ Singh, Simon (15 October 2013). "Why is the number 1,729 hidden in Futurama episodes?". BBC News Online. Retrieved 15 October 2013.
  5. ↑ Hardy, G H (1940). Ramanujan. New York: Cambridge University Press (original). p.  12.
  6. ↑ Hardy, G. H. (1921), "Srinivasa Ramanujan", Proc. London Math. Soc., s2-19 (1): xl–lviii, doi:10.1112/plms/s2-19.1.1-u The anecdote about 1729 occurs on pages lvii and lviii
  7. ↑ Marshall, Michael (24 February 2017). "A black plaque for Ramanujan, Hardy and 1,729". Good Thinking. Retrieved 7 March 2019.
  8. ↑ Sloane, N. J. A. (ed.). "SequenceA051015(Zeisel numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-02.
  9. ↑ Sloane, N. J. A. (ed.). "SequenceA005898(Centered cube numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-02.
  10. ↑ Sloane, N. J. A. (ed.). "SequenceA051624(12-gonal (or dodecagonal) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-02.
  11. ↑ Sloane, N. J. A. (ed.). "SequenceA051876(24-gonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-02.
  12. ↑ Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics, Volume 1 (3rd ed.), Springer, ISBN   0-387-20860-7 - D1 mentions the Ramanujan-Hardy number.
  13. ↑ David Mitchell (25 February 2017). "Tessellating the Ramanujan-Hardy Taxicab Number, 1729, Bedrock of Integer Sequence A198775" . Retrieved 19 July 2018.
  14. ↑ Ignacio Larrosa Cañestro (June 2016). "Relación entre las distancias de un punto D a los vértices de un triángulo equilátero, y el lado de éste" [Relationship between the distances from a point D to the vertices of an equilateral triangle, and its side.](PDF) (in Spanish). p. 5. GeoGebra zKRFfhdM.
  15. ↑ Harvey, David; Conversation, The. "We've found a quicker way to multiply really big numbers". phys.org. Retrieved 2021-11-01.
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