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Root sum equals integer (Posted on 2015-03-10) Difficulty: 3 of 5
Find all integers n for which √(25/2 + √(625/4 –n)) + √(25/2 - √(625/4 –n)) is an integer.

No Solution Yet Submitted by K Sengupta    
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Solution solution & playing around | Comment 1 of 7
The domain of f(n)=√(25/2 + √(625/4 –n)) + √(25/2 - √(625/4 –n))
Is 0≤n≤625/4=156.25
f is increasing on this domain and f(0)=5 and f(156.25)=√50
So there can be at most 3 solutions and its easy enough to find that there are only two using brute force
f(30.25)=6 (n is not an integer)

but an interesting result on squaring
(f(n))^2 = 25+2√n

This must be a perfect square.  Clearly if n=0 it's 25
it can't be 36 and if n=144 it's 49.  Beyond that at we've exceeded the domain.

  Posted by Jer on 2015-03-10 12:07:14
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