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| Single or double (Posted on 2010-12-24) |
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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.
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No Solution Yet
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Submitted by Ady TZIDON
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Rating: 4.0000 (1 votes)
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Analytic Solution
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 | Comment 2 of 6 | 
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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 Pell’s 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
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Posted by Harry
on 2010-12-24 18:54:38 |
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