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{{Short description|Hardy-Ramanujan number}} |
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__NOTOC__ |
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{{ |
{{Infobox number |
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|number = 1729 |
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{| border="1" style="float: right; border-collapse: collapse; margin-left: 15px;" |
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| divisor = 1, 7, 13, 19, 91, 133, 247, 1729 |
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|----- |
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| unicode = |
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| colspan="2" | {{Numbers_1000_-_10000}} |
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| greek prefix = |
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|----- |
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| latin prefix = |
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! colspan="2" | 1729 |
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|factorization= }} |
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|----- |
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'''1729''' is the [[natural number]] following [[1728 (number)|1728]] and preceding 1730. It is the first nontrivial [[taxicab number]], expressed as the [[Sum of two cubes|sum of two cubic numbers]] in two different ways. It is known as the '''Ramanujan number''' or '''Hardy–Ramanujan number''' after [[G. H. Hardy]] and [[Srinivasa Ramanujan]]. |
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| [[Cardinal number|Cardinal]] |
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<td>One thousand seven hundred <br />[and] twenty-nine |
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|----- |
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| [[Ordinal number|Ordinal]] || 1729th |
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|----- |
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| [[Factorization]] |
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<td><math>7 \cdot 13 \cdot 19</math> |
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|----- |
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| [[Divisor]]s || 7, 13, 19, 91, 133, 247 |
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|----- |
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| [[Roman numeral]] || MDCCXXIX |
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|----- |
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| [[Binary numeral system|Binary]] || 11011000001 |
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|----- |
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| [[Octal]] || 3301 |
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|----- |
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| [[Duodecimal]] || 1001 |
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|----- |
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| [[Hexadecimal]] || 6C1 |
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|} |
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'''1729''' is known as the '''Hardy-Ramanujan number''' after a famous anecdote of the British mathematician [[G. H. Hardy]] regarding a hospital visit to the Indian mathematician [[Srinivasa Ramanujan]]. In Hardy's words:<ref>[http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html Quotations by Hardy<!-- Bot generated title -->]</ref> |
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== As a natural number == |
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{{cquote|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 [[Interesting number paradox|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."}} |
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1729 is [[composite number|composite]], the [[squarefree]] product of three [[prime number]]s 7 × 13 × 19.{{r|sierpinski}} It has as factors 1, 7, 13, 19, 91, 133, 247, and 1729.{{r|anjema}} It is the third [[Carmichael number]],{{r|koshy}} and the first Chernick–Carmichael number.{{efn|1=It is a number in which {{harvtxt|Chernick|1939}} expressed Carmichael number as the product of three prime numbers <math> (6k+1)(12k+1)(18k+1) </math>.{{r|deza-2022|chernick|oeis-carmichael}}}} Furthermore, it is the first in the family of absolute [[Euler pseudoprime]]s, a subset of Carmichael numbers.{{r|childs}} 1729 is divisible by 19, the sum of its digits, making it a [[harshad number]] in base 10.{{r|deza-2023}} |
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1729 is the dimension of the [[Fourier transform]] on which the fastest known algorithm for multiplying two numbers is based.{{r|harvey}} This is an example of a [[galactic algorithm]].{{r|hh}} |
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The quotation is sometimes expressed using the term "positive cubes", as the admission of negative perfect cubes (the cube of a [[Negative and non-negative numbers|negative]] [[integer]]) gives the smallest solution as [[91 (number)|91]] (which is a factor of 1729): |
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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.{{r|guy}} |
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:91 = 6<sup>3</sup> + (−5)<sup>3</sup> = 4<sup>3</sup> + 3<sup>3</sup> |
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Visually, 1729 can be found in other [[figurate number]]s. 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-[[polygonal number|gonal]] and the seventh 84-gonal number.{{r|deza-deza|oeis-figurate}} |
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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". |
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== As a Ramanujan number == |
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Numbers such as |
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[[File:cube-sum-1729.png|thumb|1729 can be expressed as a sum of two positive cubes in two ways, illustrated geometrically.]] |
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{{anchor|Ramanujan number}} |
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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]] who was ill in a hospital.{{r|ew|lozano}} 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 remarked that "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways".{{r|hardy}} This conversation led to the definition of the [[taxicab number]] as the smallest integer that can be expressed as a sum of two positive [[Cube (algebra)|cubes]] in distinct ways. 1729 is the second taxicab number, expressed as <math> 1^3 + 12^3 </math> and <math> 9^3 + 10^3 </math>.{{r|lozano}} |
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1729 was later found in one of Ramanujan's notebooks dated years before the incident, and it was noted by French mathematician [[Frénicle de Bessy]] in 1657.{{r|kahle}} A commemorative plaque now appears at the site of the Ramanujan–Hardy incident, at 2 Colinette Road in [[Putney]].{{r|mm}} |
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:1729 = 1<sup>3</sup> + 12<sup>3</sup> = 9<sup>3</sup> + 10<sup>3</sup> |
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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 <math> 1 + z^3 </math>, which are also expressible as the sum of two other cubes.{{r|oa|oeis-fermat}} |
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that are the smallest number that can be expressed as the sum of two cubes in ''n'' distinct ways have been dubbed [[taxicab number]]s. 1729 is the second taxicab number (the first is 2 = 1<sup>3</sup> + 1<sup>3</sup>). The number was also found in one of Ramanujan's notebooks dated years before the incident. |
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== See also == |
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The same expression defines 1729 as the first in the sequence of "Fermat near misses" {{OEIS|id=A050794}} defined as numbers of the form 1 + ''z''<sup>3</sup> which are also expressible as the sum of two other cubes. |
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* [[1729]] |
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== Explanatory footnotes == |
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1729 is the third [[Carmichael number]] and the first absolute [[Euler pseudoprime]]. |
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{{notelist|group=alph}} |
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==References== |
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1729 is a [[Zeisel number]]. It is a [[centered cube number]], as well as a [[dodecagonal number]], a 24-[[polygonal number|gonal]] and 84-gonal number. |
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{{reflist|refs= |
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<ref name=anjema>{{cite book |
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Investigating pairs of distinct integer-valued [[quadratic form]]s 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). |
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| last = Anjema | first = Henry |
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| year = 1767 |
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| title = Table of divisors of all the natural numbers from 1. to 10000. |
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| isbn = 9781140919421 |
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| page = [https://archive.org/details/bim_eighteenth-century_table-of-divisors-of-all_anjema-henry_1767/page/(47) 47] |
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| via = the [[Internet Archive]] |
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}}</ref> |
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<ref name=chernick>{{cite journal |
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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 = 3301<sub>8</sub>, 3 + 3 + 0 + 1 = 7) and [[hexadecimal]] (1729 = 6C1<sub>16</sub>, 6 + C + 1 = 19<sub>10</sub>), but not in [[binary numeral system|binary]]. |
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| last = Chernick | first = J. |
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| year = 1939 |
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| title = On Fermat's simple theorem |
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| journal = Bulletin of the American Mathematical Society |
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| volume = 45 | issue = 4 | pages = 269–274 |
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| doi = 10.1090/S0002-9904-1939-06953-X |
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| url = https://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf |
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| doi-access = free |
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}}</ref> |
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<ref name=childs>{{cite book |
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1729 has another 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 (mathematical constant)|''e'']].<ref>[http://www.mathpages.com/home/kmath028.htm The Dullness of 1729<!-- Bot generated title -->]</ref> |
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| last = Childs | first = Lindsay N. |
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| year = 1995 |
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| title = A Concrete Introduction to Higher Algebra |
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| series = Undergraduate Texts in Mathematics |
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| url = https://books.google.com/books?id=OR5KAAAAQBAJ&pg=PA409 |
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| page = 409 |
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| publisher = Springer |
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| edition = 2nd |
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| doi = 10.1007/978-1-4419-8702-0 |
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| isbn = 978-1-4419-8702-0 |
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}}</ref> |
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<ref name=deza-2022>{{cite book |
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[[Masahiko Fujiwara]] showed that 1729 is one of four [[natural numbers]] (the others are [[81 (number)|81]] and [[1458 (number)|1458]] and the trivial case [[1 (number)|1]]) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number: |
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| last = Deza | first = Elena |
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| year = 2022 |
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| title = Mersenne Numbers And Fermat Numbers |
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| url = https://books.google.com/books?id=-Wo-EAAAQBAJ&pg=PA51 |
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| page = 51 |
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| publisher = World Scientific |
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| isbn = 978-981-12-3033-2 |
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}}</ref> |
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<ref name=deza-2023>{{cite book |
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: 1 + 7 + 2 + 9 = 19 |
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| last = Deza | first = Elena |
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: 19 · 91 = 1729 |
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| year = 2023 |
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| title = Perfect And Amicable Numbers |
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| url = https://books.google.com/books?id=qGSzEAAAQBAJ&pg=PA411 |
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| page = 411 |
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| publisher = World Scientific |
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| isbn = 978-981-12-5964-7 |
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}}</ref> |
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<ref name=deza-deza>{{cite book |
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The proof is very easy, which is probably why Fujiwara has never shown his proof. It suffices only to check sums up to 36, since an n-digit sum, when multiplied by its reversal, results in a number with at most 2n digits whose digit sum is no greater than 18n. For n > 2, 18n has less than n digits, and for n = 2, 18n = 36. In addition, since reversals and digit sums do not affect [[Modular arithmetic|mod]] 9 arithmetic, the square of the sum is congruent to the sum itself mod 9, so the sum must be congruent to 0 or 1 (mod 9) |
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| last1 = Deza | first1 = Michel-marie |
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| last2 = Deza | first2 = Elena |
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| year = 2012 |
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| title = Figurate Numbers |
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| url = https://books.google.com/books?id=ERS7CgAAQBAJ&pg=PA436 |
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| page = 436 |
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| publisher = [[World Scientific]] |
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| isbn = 978-981-4458-53-5 |
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}}</ref> |
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<ref name=ew>{{cite book |
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It has occasionally been suggested that Hardy's story is apocryphal, on the grounds that he almost certainly would have been familiar with some of these features of the number. |
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| last1 = Edward | first1 = Graham |
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| last2 = Ward | first2 = Thomas |
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| year = 2005 |
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| title = An Introduction to Number Theory |
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| url = https://books.google.com/books?id=yT3TOp-7YrQC&pg=PA117 |
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| page = 117 |
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| publisher = Springer |
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| isbn = 978-1-85233-917-3 |
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}}</ref> |
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<ref name=guy>{{cite book |
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==References to 1729== |
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| last = Guy | first = Richard K. | author-link = Richard K. Guy |
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| year = 2004 |
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| title = Unsolved Problems in Number Theory |
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| publisher = Springer |
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| edition = 3rd |
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| series = Problem Books in Mathematics, Volume 1 |
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| volume = 1 | isbn = 0-387-20860-7 |
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| doi = 10.1007/978-0-387-26677-0 |
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}} |
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<br/>{{isbn|978-0-387-26677-0}} (eBook) |
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</ref> |
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<ref name=hardy>{{cite book |
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* The television show ''[[Futurama (TV series)|Futurama]]'' contains several jokes about the Hardy-Ramanujan number. In one episode, the [[robot]] [[Bender Bending Rodriguez|Bender]] receives a Christmas card from the machine that built him labeled "Son #1729". [[Ken Keeler]], a writer on the show with a Ph. D. in Applied Math, said "that 'joke' alone is worth six years of grad school." In another episode, Bender's serial number is revealed to be the sum of two cubes: his number is 2716057 = 952<sup>3</sup> + (−951)<sup>3</sup>, while that of fellow robot Flexo is 3370318 = 119<sup>3</sup> + 119<sup>3</sup>. (This datum is one of the pieces of evidence the episode uses to establish that Bender and Flexo are a pair of good-and-evil twins.) The starship ''Nimbus'' displays the hull registry number BP-1729, which simultaneously riffs on the ''[[Starship Enterprise|USS Enterprise]]''<nowiki>'</nowiki>s NCC-1701. Finally, the episode [[The Farnsworth Parabox]] contains a montage sequence where the heroes visit several parallel universes in rapid succession, one of which is labeled "Universe 1729" (the universe where Fry, Leela and Bender are all giant rude talking [[bobbleheads]]). In the movie, "[[Bender's Big Score]]", the number of the taxi cab Fry takes home in the past is also the sum of two cubes. |
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|title = Ramanujan |
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| url = https://archive.org/details/pli.kerala.rare.37877 |
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| last = Hardy | first = G. H. | author-link = G. H. Hardy |
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|location = New York |
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| publisher = [[Cambridge University Press]] |
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| year = 1940 |
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| page = [https://archive.org/details/pli.kerala.rare.37877/page/n17 12] |
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| quote = 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." |
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}}</ref> |
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<ref name=hh>{{cite journal |
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* The physicist [[Richard Feynman]] demonstrated his abilities at [[mental calculation]] when, during a trip to [[Brazil]], he was challenged to a calculating contest against an experienced [[abacus|abacist]]. The abacist happened to challenge Feynman to compute the [[cube root]] of 1729.03; since Feynman knew that 1729 was equal to 12<sup>3</sup>+1, he was able to give an accurate value for its cube root mentally using [[interpolation]] techniques (specifically, [[binomial expansion]]). The abacist had to solve the problem by a more laborious algorithmic method, and lost the competition to Feynman. |
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| last1 = Harvey | first1 = David |
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| last2 = Hoeven | first2 = Joris van der |
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| date = March 2019 |
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| title = Integer multiplication in time <math> O(n \log n) </math> |
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| url = https://hal.archives-ouvertes.fr/hal-02070778/document |
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| journal = HAL |
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| volume = hal-02070778 |
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}}</ref> |
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<ref name=harvey>{{cite web |
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* Some reports say that the octal equivalent (3301) was the password to [[Xerox PARC]]'s main computer. |
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| last = Harvey | first = David |
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| title = We've found a quicker way to multiply really big numbers |
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| url = https://phys.org/news/2019-04-weve-quicker-big.html |
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| access-date = 2021-11-01 |
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| website = phys.org |
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| language = en |
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}}</ref> |
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<ref name=kahle>{{cite book |
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* The play ''[[Proof (play)|Proof]]'' (and its adapted [[Proof (2005 film)|Film]]) by [[David Auburn]] also contains a reference to 1729. |
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| last = Kahle | first = Reinhard |
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| year = 2018 |
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| editor-last1 = Piazza | editor-first1 = Mario |
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| editor-last2 = Pulcini | editor-first2 = Gabriele |
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| title = Truth, Existence and Explanation: FilMat 2016 Studies in the Philosophy of Mathematics |
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| series = Boston Studies in the Philosophy and History of Science |
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| volume = 334 |
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| contribution = Structure and Structures |
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| contribution-url = https://books.google.com/books?id=mM10DwAAQBAJ&pg=PA115 |
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| page = 115 |
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| doi = 10.1007/978-3-319-93342-9 |
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| isbn = 978-3-319-93342-9 |
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}}</ref> |
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<ref name=koshy>{{cite book |
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* The movie ''[[Lucky Number Slevin]]'' also references the number 1729 in association with the character Nick Fisher. |
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| last = Koshy | first = Thomas |
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| year = 2007 |
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| title = Elementary Number Theory with Applications |
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| url = https://books.google.com/books?id=d5Z5I3gnFh0C&pg=PA340 |
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| page = 340 |
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| publisher = Academic Press |
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| edition = 2nd |
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| isbn = 978-0-12-372487-8 |
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}}</ref> |
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<ref name=lozano>{{cite book |
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==Quotation== |
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| last = Lozano-Robledo | first = Álvaro |
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* "Every positive integer is one of Ramanujan's personal friends."—[[J. E. Littlewood]], on hearing of the taxicab incident. |
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| year = 2019 |
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| title = Number Theory and Geometry: An Introduction to Arithmetic Geometry |
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| url = https://books.google.com/books?id=ESiODwAAQBAJ&pg=PA413 |
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| page = 413 |
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| publisher = [[American Mathematical Society]] |
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| isbn = 978-1-4704-5016-8 |
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}}</ref> |
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<ref name=mm>{{cite web |
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==See also== |
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| last1 = Marshall | first1 = Michael |
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* [[Interesting number paradox]] |
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| title = A black plaque for Ramanujan, Hardy and 1,729 |
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* [[Berry paradox]] |
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| url = https://goodthinkingsociety.org/a-black-plaque-for-ramanujan-hardy-and-1729/ |
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* ''[[A Disappearing Number]]'', a 2007 play about Ramanujan in England during World War I. |
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| website = Good Thinking |
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| date = 24 February 2017 |
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| access-date = 7 March 2019 |
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}}</ref> |
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<ref name=oa>{{cite book |
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==References== |
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| last1 = Ono | first1 = Ken |
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| last2 = Aczel | first2 = Amir D. |
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| year = 2016 |
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| title = My Search for Ramanujan: How I Learned to Count |
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| url = https://books.google.com/books?id=JZkFDAAAQBAJ&pg=PA228 |
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| page = 228 |
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| doi = 10.1007/978-3-319-25568-2 |
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| isbn = 978-3-319-25568-2 |
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}}</ref> |
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<ref name=oeis-carmichael> |
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*[[Martin Gardner]], ''Mathematical Puzzles and Diversions'', 1959 |
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{{cite OEIS|1=A033502|2=Carmichael number of the form <math> (6k+1)(12k+1)(18k+1) </math>, where <math> (6k+1) </math>, <math> (12k+1) </math>, and <math> (18k+1) </math> are prime numbers}} |
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*[[Richard K. Guy]], ''Unsolved Problems in Number Theory'', 2nd ed., Springer, 2004. D1 mentions the Hardy-Ramanujan number. |
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</ref> |
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<ref name=oeis-fermat> |
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==Notes== |
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{{cite OEIS|1=A050794|2=Consider the Diophantine equation <math> x^3 + y^3 = z^3 + 1 </math> (<math> 1 < x < y < z </math>) or 'Fermat near misses'}} |
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{{reflist}} |
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</ref> |
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<ref name=oeis-figurate> Other sources on its figurate numbers can be found in the following: |
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==External links== |
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* {{cite OEIS|1=A005898|2=Centered cube numbers}} |
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* [http://mathworld.wolfram.com/Hardy-RamanujanNumber.html MathWorld: Hardy-Ramanujan Number] |
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* {{cite OEIS|1=A051624|2=12-gonal (or dodecagonal) numbers}} |
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* [http://www.mathpages.com/home/kmath028.htm The Dullness of 1729] |
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* {{cite OEIS|1=A051876|2=24-gonal numbers}} |
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</ref> |
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<ref name=sierpinski>{{cite book |
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[[Category:Integers|199e03 1729]] |
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| last = Sierpinski | first = W. |
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| editor-last = Schinzel | editor-first = A. |
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| year = 1998 |
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| publisher = North-Holland |
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| title = Elementary Theory of Numbers: Second English Edition |
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| url = https://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA233 |
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| page = 233 |
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| isbn = 978-0-08-096019-7 |
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}}</ref> |
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}} |
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==External links== |
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* {{mathworld|urlname=Hardy-RamanujanNumber|title=Hardy–Ramanujan Number}} |
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* {{cite web|last=Grime|first=James|title=1729: Taxi Cab Number or Hardy-Ramanujan Number|url=http://www.numberphile.com/videos/1729taxicab.html|work=Numberphile|publisher=[[Brady Haran]]|author2=Bowley, Roger|access-date=2013-04-02|archive-url=https://web.archive.org/web/20170306141337/http://numberphile.com/videos/1729taxicab.html|archive-date=2017-03-06|url-status=dead}} |
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* [http://io9.com/why-does-the-number-1729-show-up-in-so-many-futurama-ep-1445512975 Why does the number 1729 show up in so many Futurama episodes?], io9.com |
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[[Category:Integers]] |
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[[de:Tausendsiebenhundertneunundzwanzig]] |
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[[Category:Srinivasa Ramanujan]] |
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[[es:Mil setecientos veintinueve]] |
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[[fr:1729 (nombre)]] |
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[[it:1729 (numero)]] |
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[[he:1729 (מספר)]] |
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[[lv:1729 (skaitlis)]] |
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[[ml:1729 (സംഖ്യ)]] |
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[[nn:1 729]] |
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Latest revision as of 21:36, 19 December 2024
| ||||
|---|---|---|---|---|
| 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 the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It is known as the Ramanujan number or Hardy–Ramanujan number after G. H. Hardy and Srinivasa Ramanujan.
As a natural number
[edit]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 is divisible by 19, the sum of its digits, making it a harshad number in base 10.[8]
1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based.[9] This is an example of a galactic algorithm.[10]
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.[11]
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.[12][13]
As a Ramanujan number
[edit]
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 who was ill in a 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 remarked that "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways".[16] This conversation led to the definition of the taxicab number as the smallest integer that can be expressed as a sum of two positive cubes in distinct ways. 1729 is the second taxicab number, expressed as and .[15]
1729 was later found in one of Ramanujan's notebooks dated years before the incident, and it 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
[edit]Explanatory footnotes
[edit]- ^ It is a number in which Chernick (1939) expressed Carmichael number as the product of three prime numbers .[4][5][6]
References
[edit]- ^ Sierpinski, W. (1998). Schinzel, A. (ed.). Elementary Theory of Numbers: Second English Edition. North-Holland. p. 233. ISBN 978-0-08-096019-7.
- ^ Anjema, Henry (1767). Table of divisors of all the natural numbers from 1. to 10000. p. 47. ISBN 9781140919421 – via the Internet Archive.
- ^ Koshy, Thomas (2007). Elementary Number Theory with Applications (2nd ed.). Academic Press. p. 340. ISBN 978-0-12-372487-8.
- ^ Deza, Elena (2022). Mersenne Numbers And Fermat Numbers. World Scientific. p. 51. ISBN 978-981-12-3033-2.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Deza, Elena (2023). Perfect And Amicable Numbers. World Scientific. p. 411. ISBN 978-981-12-5964-7.
- ^ Harvey, David. "We've found a quicker way to multiply really big numbers". phys.org. Retrieved 2021-11-01.
- ^ Harvey, David; Hoeven, Joris van der (March 2019). "Integer multiplication in time ". HAL. hal-02070778.
- ^ 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) - ^ Deza, Michel-marie; Deza, Elena (2012). Figurate Numbers. World Scientific. p. 436. ISBN 978-981-4458-53-5.
- ^ Other sources on its figurate numbers can be found in the following:
- Sloane, N. J. A. (ed.). "Sequence A005898 (Centered cube numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Sloane, N. J. A. (ed.). "Sequence A051876 (24-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Edward, Graham; Ward, Thomas (2005). An Introduction to Number Theory. Springer. p. 117. ISBN 978-1-85233-917-3.
- ^ 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.
- ^ 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."
- ^ 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.
- ^ Marshall, Michael (24 February 2017). "A black plaque for Ramanujan, Hardy and 1,729". Good Thinking. Retrieved 7 March 2019.
- ^ 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.
- ^ 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
[edit]- Weisstein, Eric W. "Hardy–Ramanujan Number". MathWorld.
- Grime, James; Bowley, Roger. "1729: Taxi Cab Number or Hardy-Ramanujan Number". Numberphile. Brady Haran. Archived from the original on 2017-03-06. Retrieved 2013-04-02.
- Why does the number 1729 show up in so many Futurama episodes?, io9.com