Little math problems September 2018

[Albert Chan]

Using (2,2) as base, one general solution for square / cube puzzle:
Below equality work for all non-negative integer n:

(2⁶ⁿ⁺¹)² + (2⁶ⁿ⁺¹)² = 2¹²ⁿ⁺³ = (2⁴ⁿ⁺¹)³
(2⁶ⁿ⁺¹)³ + (2⁶ⁿ⁺¹)³ = 2¹⁸ⁿ⁺⁴ = (2⁹ⁿ⁺²)²

Edit: For *all* base (a,b), positive integer k, scale up k⁶ will work.

So, Valentin big examples (post #7) automatically work, because base = (2,2):

9500208482 = 2 * 41^6
993962581922 = 2 * 89^6
7081411940532321969341507101262893738689268797761826852357921359507223378 = 2 * 1234567890123^6

Looking at above k = 1234567890123, seems Valentin already know this