<s>[INST] Positive integer solutions of $a^3 + b^3 = c$ Is there any fast way to solve for positive integer solutions of $$a^3 + b^3 = c$$ knowing $c$?
My current method is checking if $c - a^3$ is a perfect cube for a range of numbers for $a$, but this takes a lot of time for larger numbers. I know $c = (a+b)(a^2-ab+b^2)$, but I can't think of a way this would speed up the process. Factoring numbers can be done fairly quickly but then the values of $a$ and $b$ have to be chosen. This is for a previous question I had about Fibonacci numbers. Any relevant literature is appreciated.  
 [/INST] $c=(a+b)(a^2−ab+b^2)$ should actually be quite helpful. After factoring $c$, you find all possibly ways of writing $c = xy$. Let $x = a + b$ or $b = x - a$, so
$$
a^2-ab+b^2 = a^2 - a(x-a) + (x-a)^2 = 3a^2 - 3ax + x^2 = y
$$
Solve this quadratic equation for a and check that a is an integer. 
Obviously many factors need not be examined: If we keep the sum $x = a+b$ fixed, then $a^3 + b^3$ is between $x^3/4$ and $x^3$, so $x$ must be between $c^{1/3}$ and $(4c)^{1/3}$. 
After reading previous posts, this seems to be a published result, cited as (Broughan, 2003). Had a look at the paper. Oh well, you can sure make this complicated. 
For not very large c, say $c < 10^9$, brute force may be quicker: Let a = 0, b = $c^{1/3}$, rounded down. As long as $a^3 + b^3 < c$ increase a by 1. If $a^3 + b^3 = c$ note that you found a solution. Then decrease b by 1, repeat until $a > b$. 
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