Series What Is the Value of Taxicab ( 6 ) ?

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(2, 3, 6) is not known; however, recent results announced by Rathbun [? ] show that Taxicab(6) is in the interval [1018, 24153319581254312065344]. In this note we show that with probability greater than 99%, Taxicab(6) = 24153319581254312065344. 1 Is 1729 a Dull Number? Srinivasa Ramanujan was one of India’s greatest, largely self-taught, mathematical geniuses. In January 1913 Ramanujan, after seeing the book Orders of Infinity, wrote to its author, the distinguished number theorist G. H. Hardy. Hardy and Littlewod studied the long list of unproved theorems which Ramanujan enclosed with his letter and sent him the letter beginning with the famous paragraph: I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes: (1) there are a number of results that are already known, or easily deducible from known theorems; (2) there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance; (3) there are results which appear to be new and important . In 1914 Ramanujan arrived in England and for a couple of years Hardy and Ramanjuan had a collaboration which led to important results. The following is a famous story told by Hardy, repeated word for word in various sources. Ramanjuan became ill in 1917 and at the age of 33 lay dying in Putney hospital. Hardy on a visit to his colleague, moved and at a loss for words, could only say “I came in a taxi 1729, that’s a pretty dull number. ” Ramanujan’s immediate rejoinder was “Oh no Hardy. 1729 is the smallest integer which can be expressed in two different ways as the sum of two cubes. ” This episode prompted Littlewood to say that “every positive integer was one of [Ramanujans’] personal friends”. The number 1729 has since become known as the Hardy-Ramanujan Number, even though this feature of 1729 was known more than 300 years before Ramanujan (more precisely, by Bernard Frénicle de Bessy in 1657, cf. [? ]). Is 1729 really a dull number? Not at all. First, 1729 is a Carmichael Number, i.e., a pseudoprime relative to every base. Also note after [? ] that, beginning at the 1729th decimal digit of the transcental number e, the next ten successive digits of e are 0719425863, in the first appearance of all ten digits in a row without repititions. Finally, are there any dull natural numbers? We can prove that there are none. Indeed, if there are dull numbers, then we can divide all numbers into two sets – interesting and dull. We then pick up the smallest dull number. Since it is the smallest uninteresting number it becomes, ipso facto, an interesting number. We must therefore remove it from the dull set and place it in the other set. But now there will be another smallest dull number. Repeating this process will make any dull number interesting. 2 How Many Taxicab Numbers Are Known? The smallest number expressible as the sum of two cubes in n different ways is called the Taxicab Number Taxicab(n). Hardy and Wright [? ] (Theorem 412) have proven that the Taxicab(n) exists for every positive integer n, but the proof is of little use in finding the number. More accurately, “1729 is the smallest integer which can be expressed as the sum of two positive cubes in two different ways. ”