The only two positive square numbers that differ by 5 are 4 and 9.  [9 and 16
differ by 7, 16 and 25 differ by 9, and the difference gets larger and larger.

So 4 is the ONLY square number which, being increased by 5, gives a square
number.

However this problem also requres that when it is diminished by 5, it also gives
a square number.  

When we diminish 4 by 5 we get -1.

Is -1 a square number?  Well yes, in a sense, because i² = -1.

So the only possible answer could be 4 which when increased by 5 gives square
number 9 and when diminished by 5, gives -1, which is the square of the
imaginary number i.


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That's just asking the same thing in another way.

5 is the only number when added to the square number 4 or subtracted from the
square number 4 to give either the square number 9, which is 3² or the square
number -1, or i², which is a square number if we consider the square of an
imaginary number as a square number.  .  

Edwin