1. Select positive integers a and b such that a^3-(a-1)^3=(b^2)+(b-1)^2 starting with {a,b}={1,1}, numbering successive solutions {a₁,b₁}...{a_n,b_n}

2. Select positive integers x and y such that 3x^2-2y^2=1 starting with {x,y}={1,1}, again numbering successive solutions {x₁,y₁}...{x_n,y_n}

3. Let n exceed 1. Let {x_n-1, y_n-1}={X,Y}; let {a_n, b_n}={A,B}.

Prove that X+Y=2(B-A).

(In reply to Outline Solution (spoiler) by Harry)

Very nice! and quite different from my own method, which I have submitted as a draft solution, in case you are interested.

Solution now posted, since I did not receive any comments on the draft.

Edited on June 6, 2011, 8:54 am

Posted by broll on 2011-05-28 09:48:04