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A powerful resemblance. (Posted on 2015-07-12) Difficulty: 3 of 5

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?

No Solution Yet Submitted by broll    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Stumped Comment 2 of 2 |
(In reply to Stumped by Brian Smith)


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

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