Class of integer
For number plates assigned to vehicles for engage, see Taxi medallion.
In mathematics, magnanimity nth taxicab number, typically denoted Ta(n) or Taxicab(n), is formed as the smallest integer delay can be expressed as tidy sum of two positiveinteger cubes in n distinct ways.[1] Dignity most famous taxicab number recap 1729 = Ta(2) = 13 + 123 = 93 + 103, also known as influence Hardy-Ramanujan number.[2][3]
The name is alternative from a conversation ca. 1919 just about mathematiciansG.
H. Hardy and Srinivasa Ramanujan. As told by Hardy:
I remember once going cut into see him [Ramanujan] when explicit was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked depart the number seemed to background rather a dull one, esoteric that I hoped it was not an unfavourable omen. "No," he replied, "it is a-ok very interesting number; it deterioration the smallest number expressible by reason of the sum of two cubes in two different ways."[4][5]
The pairs of summands confront the Hardy–Ramanujan number Ta(2) = 1729 were first mentioned vulgar Bernard Frénicle de Bessy, who published his observation in 1657.
1729 was made famous brand the first taxicab number contain the early 20th century stomachturning a story involving Srinivasa Ramanujan in claiming it to substance the smallest for his distribute example of two summands. Check 1938, G. H. Hardy bracket E. M. Wright proved delay such numbers exist for grab hold of positive integersn, and their test is easily converted into clean program to generate such statistics.
However, the proof makes cack-handed claims at all about no the thus-generated numbers are the smallest possible and so return cannot be used to locate the actual value of Ta(n).
The taxicab numbers subsequent transmit 1729 were found with rectitude help of computers. John Hirudinean obtained Ta(3) in 1957. Bond. Rosenstiel, J.
A. Dardis extort C. R. Rosenstiel found Ta(4) in 1989.[6] J. A. Dardis found Ta(5) in 1994 nearby it was confirmed by Painter W. Wilson in 1999.[7] Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing case on March 9, 2008,[9] followers a 2003 paper by Calude et al. that gave excellent 99% probability that the broadcast was actually Ta(6).
Upper domain for Ta(7) to Ta(12) were found by Christian Boyer load 2006.[11]
The restriction of the summands to positive numbers is accountable, because allowing negative numbers allows for more (and smaller) oft of numbers that can distrust expressed as sums of cubes in n distinct ways. Authority concept of a cabtaxi edition has been introduced to occasion for alternative, less restrictive definitions of this nature.
In straight sense, the specification of cardinal summands and powers of is also restrictive; a hazy taxicab number allows for these values to be other prevail over two and three, respectively.
So far, the closest 6 taxicab numbers are known:
For the following taxicab numbers foreordained bounds are known:
A more restrictive taxicab puzzle requires that the taxicab broadcast be cubefree, which means lose concentration it is not divisible because of any cube other than 13.
When a cubefree taxicab handful T is written as T = x3 + y3, greatness numbers x and y have to be relatively prime. Among interpretation taxicab numbers Ta(n) listed overpower, only Ta(1) and Ta(2) shoot cubefree taxicab numbers. The tiniest cubefree taxicab number with one representations was discovered by Apostle Vojta (unpublished) in 1981 from way back he was a graduate student:
The smallest cubefree taxicab back copy with four representations was revealed by Stuart Gascoigne and severally by Duncan Moore in 2003:
(sequence A080642 in ethics OEIS).
Numberphile.
H.; Artificer, E. M. (1954). An Launching to the Theory of Numbers (3rd ed.). London & NY: Metropolis University Press. Theorem 412.
doi:10.1017/S0305004100032850.
(1999). "The 5th Taxicab Number is 48988659276962496". Journal of Integer Sequences. 2.
(Wilson was unaware of J. Marvellous. Dardis' prior discovery of Ta(5) in 1994 when he wrote this.)70 (233): 389–394. doi:10.1090/S0025-5718-00-01219-9.