What Is the Value of Taxicab(6)? An Update
نویسندگان
چکیده
The famous story of the number 1729, the smallest integer which can be expressed as the sum of two positive cubes in two different ways, motivated the introduction of Taxicab Numbers. The smallest number expressible as the sum of two cubes in n different ways is called Taxicab(n). So, Taxicab(2) = 1729. Further on, Taxicab(5) = 48988659276962496. Computing Taxicab(n) is challenging and interesting, both from mathematical and programming points of view. The exact value of Taxicab(6) is not known; in view of the results obtained by Bernstein [1] and Rathbun [14] it follows that Taxicab(6) is in the interval [1018, 24153319581254312065344]. In [5] we proved that with probability greater than 99%, Taxicab(6) = 24153319581254312065344. In this note we improve the method used in [5] in two ways: we use (1) a larger, and (2) a better quality random sampling, namely, a sample of 562,500 quantum random integers drawn from the above mentioned interval using Quantis, [10]. As a result, we prove that the above value for Taxicab(6) is true with probability greater than 99.8%.
منابع مشابه
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)...
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