1729 (number)

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172817291730
Cardinalone thousand seven hundred and twenty-nine
Ordinal1729th
(one thousand seven hundred and twenty-ninth)
Factorization7 · 13 · 19
Divisors1, 7, 13, 19, 91, 133, 247, 1729
Roman numeralMDCCXXIX
Greek prefix,αψκθ
Binary110110000012
Ternary21010013
Quaternary1230014
Quinary234045
Senary120016
Octal33018
Duodecimal100112
Hexadecimal6C116
Vigesimal46920
Base 361C136
 
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172817291730
Cardinalone thousand seven hundred and twenty-nine
Ordinal1729th
(one thousand seven hundred and twenty-ninth)
Factorization7 · 13 · 19
Divisors1, 7, 13, 19, 91, 133, 247, 1729
Roman numeralMDCCXXIX
Greek prefix,αψκθ
Binary110110000012
Ternary21010013
Quaternary1230014
Quinary234045
Senary120016
Octal33018
Duodecimal100112
Hexadecimal6C116
Vigesimal46920
Base 361C136

1729 is the natural number following 1728 and preceding 1730. 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan. In Hardy's words:[1][2][3]

I remember once going to see him when he was ill at Putney. 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 unfavorable 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 these:

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):

91 = 63 + (−5)3 = 43 + 33

Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like −91, −189, −1729, and further negative numbers. This ambiguity is eliminated by the term "positive cubes".

Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways[4] have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657.

The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence A050794 in OEIS) defined as numbers of the form 1 + z3 which are also expressible as the sum of two other cubes.

Other properties[edit]

1729 is also the third Carmichael number and the first absolute Euler pseudoprime. It is also a sphenic number.

1729 is a Zeisel number. It is a centered cube number, as well as a dodecagonal number, a 24-gonal 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 (Guy 2004).

Because in base 10 the number 1729 is divisible by the sum of its digits, it is a Harshad number. It also has this property in octal (1729 = 33018, 3 + 3 + 0 + 1 = 7) and hexadecimal (1729 = 6C116, 6 + C + 1 = 1910), but not in binary.

1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of the transcendental number e.[5]

Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 81, 1458, and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:

1 + 7 + 2 + 9 = 19
19 × 91 = 1729

It suffices only to check sums congruent to 0 or 1 (mod 9) up to 19.

References to 1729[edit]

Quotation[edit]

See also[edit]

References[edit]

Notes[edit]

  1. ^ Quotations by Hardy
  2. ^ a b c Singh, Simon (15 October 2013). "Why is the number 1,729 hidden in Futurama episodes?". BBC News Online. Retrieved 15 October 2013. 
  3. ^ Hardy, G H (1940). Ramanujan. New York: Cambridge University Press (original). p. 12. 
  4. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 13. ISBN 978-1-84800-000-1. 
  5. ^ The Dullness of 1729
  6. ^ Bender's Big Score: Bite My Shiny Metal X documentary
  7. ^ http://www.ee.ryerson.ca/~elf/abacus/feynman.html
  8. ^ Gleick, James. The Information: A History, A Theory, A Flood. Vintage Books. New York. Ch.12. Pg.339. ISBN 978-1-4000-9623-7

External links[edit]