which is the only smallest no.that can be said to be a sum of 2 cubes in 2 diferent ways?

Answer 1

1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words [1]:
“ I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." ”

The quotation is sometimes expressed using the term "positive cubes", as the admission of negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a factor of 1729):

91 = 63 + (−5)3 = 43 + 33

Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like −91, −189, −1729, and further negative numbers. This ambiguity is eliminated by the term "positive cubes".

Numbers such as

1729 = 13 + 123 = 93 + 103

that are the smallest number that can be expressed as the sum of two cubes in n distinct ways have been dubbed taxicab numbers. 1729 is the second taxicab number (the first is 2 = 13 + 13). The number was also found in one of Ramanujan's notebooks dated years before the incident.

1729 is the third Carmichael number and the first absolute Euler pseudoprime.

Answer 2

The Number = 1729

1729 can be expressed as the cubes of two numbers as shown below:

(1^ 3) + (12 ^ 3) = 1729

(9 ^ 3) + (10 ^ 3) = 1729.

So 1729 is the only smallest number that can be expressed as the sum of the cubes of two different pairs of numbers.

One of the greatesrt ever Mathematicians Srinivasa Ramanujam told Hardy the special significance of this number.

Quite amazing that numbers spoke to Ramanujam!

Another signiificance of this number 1729 is it involves the cubes of the lowest one-digit number 1 & the highest one-digit number 9 .

Answer 3

1729 also known as the Ramanujan number is the answer.
12*12*12=1728,1*1*1=1
10*10*10=1000,9*9*9=729

Adding these we get the answer

Answer 4

zero can be expressed as n^3 + (-n)^3 for all n. Whether there is a smaller number than zero satisfying the requirement, I am unsure.