Is there any fast way to solve for positive integer solutions of a3+b3=c knowing c?
My current method is checking if c−a3 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)(a2−ab+b2), 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.
Answer
A very fast way to see that a positive integer c is not the sum of two cubes are modular constraints. The equation a3+b3=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.
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