Curious properties sometimes lurk within seemingly undistinguished numbers.
Consider the story concerning Indian mathematician Srinivasa Ramanujan (1887–1920). His friend G.H. Hardy (1877–1947) once remarked that the taxi by which he had arrived had a "dull" number–1729, or 7 x 13 x 19. Ramanujan was quick to point out that 1729 is actually a "very interesting" number. It's the smallest whole number expressible as a sum of two cubes in two ways: Both 13 + 123 and 93 + 103 equal 1729.
The first published reference to this property of the integer 1729 is in the writings of 17th-century French mathematician Bernard Frénicle de Bessy (1605–1670).
You might then wonder about the identity of the smallest number expressible as the sum of two cubes in three different ways. It's 87539319, discovered in 1957 by John Leech (1926–1992) in the course of an extensive computer search.
87539319 = 1673 + 4363 = 2283 + 4233 = 2553 + 4143
Nowadays, mathematicians define the smallest number expressible as the sum of two cubes in n different ways as the nth taxicab number, denoted Taxicab(n). Hence, Taxicab(2) = 1729 and Taxicab(3) = 87539319.
Interestingly, Hardy and E.M. Wright had proved a theorem guaranteeing that the taxicab number exists for any value of n greater than or equal to 1. So the search was on, but finding the numbers turned out to be exceedingly difficult.
Taxicab(4) was discovered in 1991 by amateur number theorist E. Rosenstiel, who obtained expert help from computer scientists J.A. Dardis and Colin R. Rosenstiel to track it down.
Taxicab(4) = 6963472309248 = 24213 + 190833 = 54363 + 189483 = 102003 + 180723 = 133223 + 166303
David W. Wilson found the fifth taxicab number on Nov. 21, 1997. A few months later, Daniel J. Bernstein came upon the same number in an independent investigation.
Taxicab(5) = 48988659276962496 = 387873 + 3657573 = 1078393 + 3627533 = 2052923 + 3429523 = 2214243 + 3365883 = 2315183 + 3319543
Wilson's discovery had occurred during a lengthy computer search in which he was trying to extend the list of known pairs of cubes that add up to the same number in four ways. The same search turned up other examples of five-way sums of two cubes and 8230545258248091551205288 as the smallest known six-way sum of cubes.
In 1998, Bernstein found a smaller example of a six-way sum of cubes. He also established that the sixth taxicab number had to be greater than 1018, fencing in the possible value of Taxicab(6).
Recently, Randall L. Rathbun came upon an even smaller candidate: 24153319581254312065344. Does it qualify to be Taxicab(6)? No one knows yet. A smaller candidate for the least six-way sum of cubes might lurk among the numbers between 1018 and Rathbun's result.
It's startling to note that very little is yet known about taxicab numbers. Some related questions are even more challenging. For example, you can ask, as Hardy did of Ramanujan, for the smallest number that is a sum of two fourth powers in two ways. Ramanujan couldn't provide an immediate answer, but it was known to Leonhard Euler (1707–1783): 635318657 = 594 + 1584 = 1334 + 1344.
Then it gets tougher. The first number representable as the sum of two fourth powers in three ways must, if exists, have at least 19 digits. Unlike the situation for cubes, however, there is no theorem yet that guarantees the existence of the relevant fourth-power sums.
Are there any numbers that are a sum of two fifth powers in two ways? No one appears to know.
You can also look for the equivalent of taxicab numbers when you allow both positive and negative cubes. For example, in the case of three-way sums of cubes, 728 = 63 + 83 = 93 – 13 = 123 – 103. The smallest positive integer that can be written as the sum of two positive or negative cubes in n ways is sometimes called the nth cabtaxi number. The eighth cabtaxi number is now known, and the ninth must have at least 19 digits.
In this nook of number theory, as in many others, questions abound and answers are elusive. In 1991, Rosentiel and his collaborators concluded, "This leaves open the possibility of further projects, as and when yet more powerful computers with suitable algorithms can be applied to these remaining unsolved problems." More than a decade later, progress remains slow.