Trisha Shetty (Editor)

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.

T a x i c a b ( 1 , 2 , 2 ) = 4 = 1 + 3 = 2 + 2. T a x i c a b ( 2 , 2 , 2 ) = 50 = 1 2 + 7 2 = 5 2 + 5 2 . T a x i c a b ( 3 , 2 , 2 ) = 1729 = 1 3 + 12 3 = 9 3 + 10 3 - famously stated by Ramanujan.

Euler showed that

T a x i c a b ( 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.

References

Generalized taxicab number Wikipedia
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