Question

find all integers of n such that :
sqrt(50+sqrt(n)) + sqrt(50-sqrt(n)) is an integer number.

Answer 1

\[\sqrt{50+\sqrt{n}} + \sqrt{50-\sqrt{n}} = k\]

Answer 2

For the given expression to be an integer, it must be the case that both \(50+\sqrt{n}\) and \(50-\sqrt{n}\) are perfect squares :
\[50+\sqrt{n}=a^2\\50-\sqrt{n}=b^2\\a\ge b\]
add and get
\[100=a^2+b^2\]
Clearly this can be done in exactly two ways : \((a,b)=(10,0),~(8,6)\) and the corresponding \(n\) values are \(n=2500,~196.\)

Answer 3

I have to log in to give you medal. Admire!!!

Answer 4

you just miss one solution again @rational , 2016 is works also

Answer 5

wow! 2016 is also working, how come i missed that

Answer 6

yeah there are 3 solutions for n :D
n = 196, 2500, 2016

Answer 7

maybe we can use these formula :
\[\sqrt{a+b + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b} \]

Answer 8

and \[\sqrt{a+b - 2\sqrt{ab}} = \sqrt{a} - \sqrt{b}\]

Answer 9

with a >= b