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Surprising Sudden Square (Posted on 2007-03-24) Difficulty: 3 of 5
Prove that if a²+b² is a multiple of ab+1, for positive integer a and b, then (a²+b²)/(ab+1) is a perfect square.

No Solution Yet Submitted by Old Original Oskar!    
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Solution chains | Comment 6 of 10 |

Let (a, b) be any solution pair with a>b.  Let s = (aČ+bČ)/(ab+1).  Then another solution can be derived by creating solution pair (s*a-b, a).

Plugging in the trivial solution of (a, 0) with a>1 and repeatedly applying the transform above creates the following sequence:
(a, 0)
(a^3, a)
(a^5-a, a^3)
(a^7-2a^3, a^5-a)
(a^9-3a^5+a, a^7-2a^3)
(a^11-4a^7+3a^3, a^9-3a^5+a)
(a^13-5a^9+6a^5-a, a^11-4a^7+3a^3)
(a^15-6a^11+10a^7-4a^3, a^13-5a^9+6a^5-a)

For different choices of a, I believe this algorithm will generate every solution.

  Posted by Brian Smith on 2007-03-25 12:02:56
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