Find all the quadruples [a, b, c, d] where a^3 + b^3 = c^3 + d^3 when 1 <= a, b, c or d <= 10000 [closed]

Answer 1

Let's assume a solution:

a=A, b=B, c=C, and d=D.

Given any solution we can generate another 3 solutions

abcd
ABCD

ABDC
BACD
BADC

Actually, if A=B, or C=D, then we might only have 1 or 2 further solutions.

We can choose the solutions we look for first by ordering A <= B and C <= D. This will reduce the search space. We can generate the missed solutions from the found ones.

There will always be at least one solution, where A=C and B=D. What we're looking for is when A>C and B. This comes from the ordering: C can't be greater than A because, as we've chosen to only look at solutions where D>C, the cube sum would be too big.

We can calculate A^3 + B^3, put it in a map as the key, with a vector of pairs A,B as the value.

There will be (n^2)/2 values.

If there are already values in the vector they will all have lower A and they are the solutions we're looking for. We can output them immediately, along with their permutations.

I'm not sure about complexity.