SOLUTION: How can i find all the positive integers "n" so that 2^17 + 17 * 2^12 + 2^n is a perfect square?

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Question 557998: How can i find all the positive integers "n" so that
2^17 + 17 * 2^12 + 2^n is a perfect square?

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The expression factors to

for some integer k. Hence,



Both k - 448 and k + 448 have to be powers of 2. They differ by 896 (128*7), and it can easily be checked that {128, 1024} is the only pair of powers of 2 differing by 896 (to prove this, let m > n, 2^m - 2^n = 128*7, factor, etc). Hence, k - 448 = 128 and k + 448 = 1024 so k = 576. Hence,



n = 17 is the only positive integer satisfying the condition.