What is the Value of Taxicab(6)?

Abstract

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 Taxicab 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 [1¹⁸, 24153319581254312065344]. In this note we show that with probability greater than 99%, Taxicab(6) = 24153319581254312065344.