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Single or double (Posted on 2010-12-24) Difficulty: 3 of 5
Both a number N and its half,when augmented by 1 (i.e.N+1 and N/2+1) form a perfect square.
0 is a trivial example.

List 4 more examples.

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)

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Solution Analytic Solution | Comment 2 of 6 |
Let        N + 1 = a2         and       N/2 + 1 = b2      so that the integers a and b

satisfy the equation:       a2 - 2b2 = -1

This is a form of Pellís equation, and the values of a and b can be found by writing sqrt(2) as a continued fraction and using appropriate convergents as follows.

Convergents of sqrt(2):              1/1, 3/2, 7/5, 17/12, 41/29, 99/70, ...

Alternate convergents give possible pairs (a, b) as: (1, 1), (7, 5), (41, 29),...

In fact, successive values of a and b can be found using the same recurrence rule:

an = 6*an - 1 - an - 2           and       bn = 6*bn - 1 - bn - 2

Here are a few:              n          a                      b                      N
                                    1          1                      1                      0
                                    2          7                      5                      48
                                    3          41                     29                     1680
                                    4          239                   169                   57120
                                    5          1393                 985                   1940448



  Posted by Harry on 2010-12-24 18:54:38
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