Special numbers 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 ?
 A: 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.
A: (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.
