'Pseudocubality' -- fast tests for large possible-cubes I'm looking for a way to quickly show that a huge number is not a cube. The numbers are of the form a^b + c^d where a and c are less than a million and b and d are under a hundred (give or take on both). a and c are also known to be odd, and the exponents are at least 3. At the moment I'm checking a+c mod 8 (it must be 0) and the sum of the powers, calculated with fast modular exponentiation, mod 9, 13, and 19. (I'm also checking mod 7, but this doesn't seem to help -- any ideas on why that is the case?) What I want are more tests, since far too many non-cubes slip through these tests. Since most of the inputs will be handled by one of these, the test doesn't have to be blazingly fast -- although faster is better, to be sure. Is there a cubic reciprocity test that can practically handle numbers of this size? Can it be done (like modular exponentiation) without expanding the numbers?