Problem 3

Determine the maximum value of

m^2+n^2

, where

m

and

n

are integers satisfying

m, n \in \{1, 2, \ldots, 1981\}

and

(n^2-mn-m^2)^2 = 1

Step 4 of 6: Conclude every solution is a Fibonacci pair

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\text{All positive solutions are } (m,n) = (F_k, F_{k+1})

Detailed analysis

Since every solution descends (Step 3) to the base case and every step of the forward recurrence (Step 2) preserves being a solution, by induction every positive integer solution $(m,n)$ with $n\ge m$ is a pair of consecutive Fibonacci numbers $(F_k,F_{k+1})$ with $F_1=F_2=1$ .