Number Theory Taxicab Number

Question

How to prove that there are infinite taxicab numbers? ok i was reading this http://en.wikipedia.org/wiki/Taxicab_number#Known_taxicab_numbers and thought of this question..any ideas?

Answer 1

It is easy to show that there are infinitely many positive integers which are representable as the sum of two cubes, e.g., see the article Characterizing the Sum of Two Cubes by K.A. Broughan (2003). If we require a representation as the sum of two cubes in at least $N\ge 2$ different ways, then the result is more difficult to show; and the proof uses the theory of elliptic curves etc. For a good survey, see the article Taxicabs and sum of two cubes by J. H. Silverman. In particular, the following result due to K. Mahler is discussed:

Theorem(Mahler): There is a constant $c>0$ such that for infinitely many positive integers $m$, the number of positive integer solutions to the equation $x^3+y^3=m$ exceeds $c(\log(m))^{1/3}$.

Answer 2

Use Ramanujan's identity that $$\left(x^2+7xy-9y^2\right)^3+\left(2x^2-4xy+12y^2\right)^3=\left(2x^2+10y^2\right)^3+\left(x^2-9xy-y^2\right)^3$$

Reference:

http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html