IIT JEE , AIEEE Forum - Mathematics , Algebra

elastiboysai

Determine the maximum value of m² + n², where m and n are integers in the range 1, 2, ... , 1981 satisfying

(n² - mn - m²)² = 1.

netkid07

is the answer.....3524578 ???

elastiboysai

Edited

I think i can wait for the soln

netkid07

put small values and u'll notice that the solutions of..........

n^2 - mn - m^2 = 1 or -1..........are succesive fibonacci numbers

so, let's suppose n>m is a solution

this suggests trying m+n, n: (m+n)^2 - (m+n)n - n^2 = m^2 + mn - n^2 = -(n^2 - mn - m^2) =

1 or -1. So if n > m is a solution, then m+n, n is another solution..............

starting from 2,1 it gives 3,2; 5,3; 8,5; 13,8; 21,13; 34,21; 55,34; 89,55; 144,89; 233,144;

377,233; 610,377; 987,610; 1597,987; 2584,1597

now, we need to know if there are more solutions or not !!

we try......., n-m: m^2 - m(n-m) - (n-m)^2 = m^2 + mn - n^2 = -(n^2 - mn - m^2) = 1 or -1

so this also satisifies the equation........if m>1, then m>n-m (if not then n>=2m so

n(n-m)>=2m^2 , so n^2-nm-m^2 > m^2 > 1 )

so given a solution n > m with m > 1, we have a smaller solution m > n-m this process

must finish at a solution n, 1 with n>1. but such solution is (2,1).......hence the starting solution must have been in the forward sequence from (2,1)

hence the solution is 1597^2 + 987^2=2550409 + 974169 = 3524578.........

hope you get it........cheers !

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