IIT JEE , AIEEE Forum - Mathematics , Algebra - Solve in positive integers m,n the equation

hpudipeddi

2^3m+1 + 3²ⁿ + 5^m + 5ⁿ = 2003

me

we have --------
2.8^m + 9^n + 5^m + 5^n = 2003
as
9^4 > 2003 & 2.8^4 > 2003
hence 3>=n & 3>=m
also n=3 & m=3 is clearly a solution.

now observe that max value of 2.8^m + 5^m is 1149 (as 3>=m)

now if 3>n (n not= 3) then value of 9^n + 5^n is less than 854.

and net sum is less than 2003

thus we get 3 not> n.

similarly 3 not> m

we gat only solution as --------
n=m=3.

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