|
In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number which can be expressed as the sum of j kth positive powers in n different ways. For k = 3 and j = 2, they coincide with Taxicab numbers. \mathrm{Taxicab}(1, 2, 2) = 4 = 1 + 3 = 2 + 2. \mathrm{Taxicab}(2, 2, 2) = 50 = 1^2 + 7^2 = 5^2 + 5^2. \mathrm{Taxicab}(3, 2, 2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3 - famously stated by Ramanujan. It has been shown by Euler that \mathrm{Taxicab}(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4. However, Taxicab(5, 2, n) is not known for any n ≥ 2; no positive integer is known which can be written as the sum of two fifth powers in more than one way.[1] It can be easily verified on a home computer using a simple brute force search that the Taxicab(5, 2, 2) problem has no solutions with { a, b, c, d } all less than 1,000. A more in-depth search shows the same is true for all combinations up to 4,000. A lower bound on the solution is |
About us|Jobs|Help|Disclaimer|Advertising services|Contact us|Sign in|Website map|Search|
GMT+8, 2015-9-11 21:24 , Processed in 0.134684 second(s), 16 queries .
