Generalized taxicab number
Encyclopedia
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 number
s.
It has been shown by Euler that
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 number
Taxicab number
In mathematics, the nth taxicab number, typically denoted Ta or Taxicab, is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways. The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th...
s.
It has been shown by Euler that
However, Taxicab(5, 2, n) is not known for any n ≥ 2; no positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
is known which can be written as the sum of two fifth powers in more than one way.
External links
- Walter Schneider:Taxicab numbers