Special numbers

Question

Our teacher talked about some special numbers.

These numbers total of 2 different numbers' cube. For example :

$x^3+y^3 = z^3+t^3 = \text{A-special-number}$

What is the name of this special numbers ?

Answer 1

Apparently you were looking for the taxicab numbers whose name derives from an anecdote of G.H. Hardy on his visiting Ramanujan:

I remember once going to see him [Ramanujan] 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."

Copied from the Wikipedia page on 1729.

There are various incarnations of the taxicab numbers

A001235: Taxi-cab numbers: sums of 2 cubes in more than 1 way:

1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 
110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 
262656, 314496, 320264, 327763, 373464, 402597, 439101, 443889, 513000, 
513856 

and J.M. linked to

A011541 : Taxi-cab (taxicab) or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 cubes in n ways (an infinite sequence):

2, 1729, 87539319, 6963472309248, 48988659276962496, 24153319581254312065344

And here's Durangobill's page on Ramanujan numbers which I found by Googling.

To finish this collection of links, let me quote J.E. Littlewood:

Every positive integer is one of Ramanujan's personal friends.

(again from the Wikipedia page on 1729.)

Answer 2

Here's a sample identity by Ramanujan,

$(3x^2+5xy-5y^2)^3 + (4x^2-4xy+6y^2)^3 = (-5x^2+5xy+3y^2)^3 + (6x^2-4xy+4y^2)^3$

Let {x,y} = {-1,0} and you get the nice $3^3+4^3+5^3 = 6^3$. Or {x,y} = {-1,2} for $1^3+12^3 = 9^3+10^3 = 1729$, the smallest non-trivial "taxicab number" (after transposition and removing common factors). And so on.

Answer 3

(This is a reply to Eray’s question about the 2nd smallest taxicab number 4104.) Ramanujan’s formula in quadratic polynomials is not complete. You need Binet’s formula in 4th deg polynomials to answer your question, namely,

$((1-m(p-3q))r)^3 + ((-1+m(p+3q))r)^3 + ((m^2 -(p+3q))r)^3 + ((-m^2 +(p-3q))r)^3 = 0$

where,

$m = p^2+3q^2$

For any given non-trivial solution to $a^3+b^3+c^3+d^3 = 0$, you can always find its rational {m,p,q,r}. For example, let {m,p,q,r} = {3/4, 3/4, -1/4, -16}, then you will find this yields the second smallest taxicab number,

$2^3+16^3 = 9^3+15^3 = 4104$

For more details how I found {m,p,q,r}, see the short discussion on Binet’s formula in Form 2 of http://sites.google.com/site/tpiezas/010.