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Determine analytically the maximum of a²+b² if a and b belong to the set {1,2...10000} and satisfy |a²-ab-b²|=1.
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Submitted by atheron
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Rating: 5.0000 (1 votes)
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Solution:
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Consider a pair (n,m) that satisfies the problem. If m>1 then n>m if m=1 then n=1 or n=2. If p=n-m then 1=(n^2-nm-m^2)^2=(m^2-pm-p^2)^2.
This means that pair (m,n-m) also satisfies the problem. If n-m>1 we can do the same thing and get a sequence of pairs satisfying the problem. n=n_k m=n_k-1... The sequence stops when n_i=1 which means that n_i-1=2. Therefore the sequnce is the set of Fibonacci numbers backwards. This means that Fibonacci numbers satisfy the given equation.
Since 6765 is the greatest Fibonacci number less than 10000, pair (6765,4181) maximizes the sum of the squares. |
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