How can I prove that the number 1729 is a pseudoprime ? I know that a pseudoprime is a composite n where n|((2^n)-2).
As you already wrote, you only need to evaluate $\displaystyle 2^{1729} \mod 1729$ (basically, verify that 1729 fulfills the definition).
You should be familiar with [ur=http://en.wikipedia.org/wiki/Modular_arithmetic]modular arithmetics[/url] and congruences to do this.
Whether you want to this by hand or by computer, an effective way is to compute powers for the exponents:
1729 -> 864 -> 432 -> 216 -> 108 -> 54 -> 27 -> 13 -> 6 -> 3 -> 1
(If you already know n-th power, then you can easily compute (2n)-th power, and by multiplying by 2, you get the (2n+1)-st power.)
This method is described in this wikipedia article.
For instance, you can start as
$\displaystyle 2^6 \equiv 64 \pmod{1729}$
$\displaystyle 2^{12} \equiv 64^2 = 2\cdot 2048 \equiv 2\cdot319 \equiv 638
\pmod{1729}$
$\displaystyle 2^{13}\equiv 2\cdot638 = 1276 \equiv -453 \pmod{1729}$
$\displaystyle 2^{26} \equiv 453^2 \equiv \dots \pmod{1729}$
etc.