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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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re(2): solution & playing around | Comment 4 of 7 |
(In reply to re: solution & playing around by Charlie)

generalizing this result we have that 


n=4k^4+8k^3-44k^2-48k+144 for integer k>=1
gives all values of n for which f(n) is an integer

this can be derived from the result of:
f(n)^2=25+2sqrt(n)
thus if f(n)=t then
t^2=25+2sqrt(n)
25-t^2=2sqrt(n)
(25-t^2)^2=4n
n=(25-t^2)^2/4

25 = 1 mod 4
if t is odd then t^2 = 1 mod 4 and thus 25-t^2 = 0 mod 4

thus we simply need that k be an odd integer
substituting t=2k+1 we get
n=(25-(2k+1)^2)^2/4
n=4k^4+8k^3-44k^2-48k+144

  Posted by Daniel on 2015-03-10 15:13:01
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