Little math problems September 2018

[Albert Chan]

(09-20-2018 02:37 PM)Albert Chan Wrote:  Edit: found the general form of sum of 2 squares a cube. Not sure if it cover all cases ...

It only take 1 counter-example to dis-prove a conjecture ...

3430000^2 + 10290000^2 = 49000^3 proved the pattern (2 N^3, 2 N^3), (N^3, 2 N^3) were wrong.

This counter-example also proved above link does not cover all cases for two squares sum to a cube:

(A^3 - 3 A B^2)^2 + (3 A^2 B - B^3)^2 = (A^2 + B^2)^3

Searching all integers A, B, 0 <= A <= B, and A^2 + B^2 = 49000, we have 2 solutions.

A, B = 70 , 210 ---> 6174000^2 + 8918000^2 = 49000^3
A, B = 126, 182 --> 2639728^2 + 10520496^2 = 49000^3

Since it is unable to generate the counter-example, the formula does not cover all cases.