Is it too easy or too tough?
Let and be positive integers satisfying , find .
The answer is 59.
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1 solution
x^2 = 2^n - 615
Then for some number i, [2^(n/2)-i]^2 must be a perfect square.
So i^2 - 2^(n/2+1)*i + 615 = 0
To factor this for an integer solution, two factors of 615 must sum to a power of 2.
The factors of 615 are (3,205), (5,123), and (15, 41). Of these only the second sums to a power of two: 5+123 = 128.
Then 2^(n/2+1)=128, so that n=12, and x=59.
My apologies--for some reason I couldn't get the LaTex editor to work.