1. Let a and b be consecutive non-zero whole numbers. Let C be the square of their sum, less their product, i.e. C = (a+b)^2-ab. Let n be a whole number.

Claim 1: C^n = (A+B)^2-AB, for some whole numbers A,B.

Prove it, or find a counter-example.

2. Let a and b be ANY relatively prime whole numbers, and perform the same operation.

Claim 2: C^n = (A+B)^2-AB, for some whole numbers A,B.

Any exceptions?

Brian,

As I remember, all these ideas cropped up at about the same time - the trigger was one of KS's problems about cubes and 5th powers, I think.

And you are right, there is a connection with  Searching for c among the cubes.

Edited on January 1, 2017, 11:24 pm

Posted by broll on 2017-01-01 23:15:58