Generalized taxicab number
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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.
It has been shown by Euler that
- Taxicab(4,2,2) = 635318657 = 594 + 1584 = 1334 + 1344
However, Taxicab(4, 3, n) is not known for any n ≥ 2, and neither is Taxicab(5, 2, n); in fact, no positive integer is known at all which can be written as the sum of three fourth powers or two fifth powers in more than one way.
Recently, three students at the University of Michigan, Sam Rosenbaum, Penn Chou, and Will Buckingham, discovered a number which can be written as the sum of three fourth powers in two ways. It may not be the first such number, but it proves that there is such a number. This number is 202,698,520,363,438,593 = 3,4814 + 9,3224 + 21,0144 = 7,9064 + 17,6894 + 17,8224
