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.
[Math] Positive integer solutions of $a^3 + b^3 = c$
diophantine equationsnumber theorypolynomials
Best Answer
A very fast way to see that a positive integer $c$ is not the sum of two cubes are modular constraints. The equation $a^3+b^3=c$ has no integer solutions if $c$ satisfies one of the following congruences:
$$ (1) \quad c\equiv 3,4 \mod 7 $$
$$ (2) \quad c\equiv 3,4,5,6 \mod 9 $$
$$ (3) \quad c\equiv 3,4,5,6,10,11,12,13,14,15,17,18,21, 22, 23, 24, 25, 30, 31, 32, 33, 38, 39, 40, 41, 42, 45, 46, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60 \mod 63 $$
On the other hand, there are intrinsic properties of $c$ known, such that $c$ is the sum of two squares:
Theorem (Broughan 2003): Let $c$ be a positive integer. Then the equation $c=a^3+b^3$ has solutions in positive integers if and only if the following conditions are satisfied:
1.) There exists a divisor $d\mid c$ with $c^{1/3}\le d\le 2^{2/3}c^{1/3}$ such that
2.) for some positive integer $l$ we have $d^2-c/d=3l$ such that
3.) the integer $d^2-4l$ is a perfect square.