Algorithm to find the numbers expressible as the sum of two positive cubes in two different ways I have known this from beginning that $1729$ is the smallest number expressible as the sum of two cubes in two different ways: 
$$ 12^3 + 1^3 $$
and 
$$ 10^3+9^3 $$
I am a Software Developer and if someone can tell me the logic to write a program for printing such types of number will be greatly helpful.
 A: This would be essentially solutions to the Diophantine equation, $a^3+b^3=c^3+d^3.$ Which I believe Euler solved for all rational numbers with :$(3a^2+5ab−5b^2)^3+(4a^2−4ab+6b^2)^3+(5a^2−5ab−3b^2)^3 = (6a^2−4ab+4b^2)^3$
This can be rewritten as 
$(A^2+7AB−9B^2)^3+(2A^2−4AB+12B^2)^3 = (2A^2+10B^2)^3+(A^2−9AB−B^2)^3$
with $a=A+B, b=A-2B.$
The solutions to all integer values only I don't believe has been found, but the solution to some integer values at least has been, and can be found in this paper by Marc Chamberland from 1999:
http://www.fq.math.ca/Scanned/38-3/chamberland.pdf
(This is the first answer I posted in this site so I'm not sure if my format for posting is correct, apologies if not)
A: The bruteforce approach is simple.
Loop over integers $k$.
For $1\leq i<\frac k2$ check whether or not $i$ has a cubic root, and whether or not $k-i$ has a cubic root. If so, collect the pair.
When the collection of pairs has more than one pair, collect $k$.
Stop at some arbitrarily large integer, and print the collected $k$'s and the pairs collected for each.
There is probably a much better approach using heuristics though.
A: One logic to generate some taxi-cab numbers is to consider the equation 
$$1729=11^3+1^3=10^3+9^3$$
Now multiply this equation by $k^3$, this gives you $$1729k^3=(11k)^3+(k)^3=(10k)^3+(9k)^3.$$ 
Now put $k=1,2,3,4,....$ to obtain some terms of this sequence.
