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the number 1729

Since then the number 1729 has been known as the taxicab number and is denoted as denoted Ta(n) or Taxicab(n) in mathematics.

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the smallest integer that can be expressed as a sum of two positive third powers in n distinct ways

A taxicab number is the smallest integer that can be expressed as a sum of two positive third powers in n distinct ways.

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as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia)

The nth taxicab number, denoted as T(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia).

Numbers that can be expressed as the sum of two cubes in two different ways

Numbers that can be expressed as the sum of two cubes in two different ways are referred to as Taxicab numbers or Hardy-Ramanujan numbers.

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as the smallest number that can be expressed as a sum of a number of powers in $n $ different ways and is denoted as

Extending this concept a little further, a generalized taxicab number can be defined as the smallest number that can be expressed as a sum of a number of powers in $n $ different ways and is denoted as .

the smallest number which can be expressed as the sum of j kth positive powers in n different ways

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.

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the positive numbers representable in minimum 2 ways as a sum of positive cubes

Taxicab numbers are the positive numbers representable in minimum 2 ways as a sum of positive cubes.

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a number that is the first number expressible as a sum of two (positive) cubes in x different ways

While a taxicab number is a number that is the first number expressible as a sum of two (positive) cubes in x different ways, a cabtaxi number is the first number expressible as a sum of two positiive, zero, or negative cubes in x different ways -in other words, the sum of difference of two positive, zero, or negative cubes.

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on the taxis

Each taxicab has a unique number on the taxis.

the fifth taxicab number

Mathematics: 48,988,659,276,962,496 is the fifth taxicab number.

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