Power problem
Number Theory
Level
4
Let the number of natural numbers that satisfy the condition: 2^8+2^11+2^n= a perfect square be k.Let the sum of all solutions be l. find the value of k+l.
The answer is 13.
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1 solution
It can be shown there exists no solution of n<=8. We can write the given expression as: 2^8(9+2^(n-8)). This is supposed to be a perfect square. Hence,9+2^(n-8) should be a perfect square.Let it be x^2. 9+2^(n-8)=x^2 => 2^(n-8)=x^2-9=(x-3)(x+3) Each of (x-3) and (x+3) should thus be powers of 2. This is possible only for x=5 and n=12. Thus,k=1. The only solution for this is n=12. Hence,l=12. Thus,k+l=13.