Cristian Calude, Elena Calude, Michael J. Dinneen. What is the Value of Taxicab(6)?. J. UCS, 9(10):1196-1203, 2003. [doi]
For almost 350 years it was known that 1729 is the smallest integer which can be expressed as the sum of two positive cubes in two different ways. Motivated by a famous story involving Hardy and Ramanujan, a class of numbers called Taxi-cab Numbers has been defined: Taxicab(k,j,n) is the smallest number which can be expressed as the sum of j kth powers in n different ways. So, Taxicab(3,2,2) = 1729;Taxicab(4,2,2) = 635318657. Computing Taxicab Numbers is challenging and interesting, both from mathematical and programming points of view. The exact value of Taxicab(6) = Taxicab(3, 2, 6) is not known; however, recent results announced by Rathbun [R2002] show that Taxicab(6) is in the interval [1018,24153319581254312065344].