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

<math>Taxicab(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4</math>

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.

External links

• Walter Schneider: Taxicab numbersfr:Nombre taxicab généralisé