# 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 ?

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'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."' - G.H. Hardy, copied from here. –  commenter Sep 25 '11 at 11:26
Ah yeah, the taxicab numbers. –  Ｊ. Ｍ. Sep 25 '11 at 11:30
@commenter yes! exactly this ! thank you. 1729 is first of Ramanujan numbers. So I need 2nd Ramanujan number. Did you know it? It's between 4000 and 5000 –  Eray Sep 25 '11 at 11:33
The next ones are 1729, 4104, 13832, 20683, 32832 according to the list from OEIS and here's a further link. –  commenter Sep 25 '11 at 11:38
Thank you commenter and J.M. . @commenter, can you send these as an answer? So i can accept it? :) –  Eray Sep 25 '11 at 11:42

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."

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


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.

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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.

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thank you. This will help me really. But can you explain it more detailly please? –  Eray Sep 28 '11 at 22:16
Dear Eray. This is as simple as it gets. Let x = -1, and y = 0, and the identity will give you 3^3+4^3 = (-5)^3+6^3. Of course, the identity is true for any {x,y}. –  Tito Piezas III Oct 4 '11 at 9:50
for example which {x,y} pair equals to 4104 ? (4104 is 2nd smallest number of taxicab numbers) –  Eray Oct 6 '11 at 11:05

(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.

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