SOLUTION: Find a square number such that, when five is added or subtracted, the result is again a square number.

Let the square number be a²

Let b² be the square number when 5 is added. So a² + 5 = b²

Let c² be the square number when 5 is subtracted. So a² - 5 = c²

Add the two equations term by term: 

Multiply the 2nd equation by -1 and add them term by term:

 a² + 5 =  b²
-a² + 5 = -c²
-------------
     10 = b²-c²

     10 = (b-c)(b+c)

b-c is smaller than b+c
10 can only be factored as 1∙10 and 2∙5

Trying b-c = 1 and b+c = 10



Solve the first equation for b

b = 1+c

Substituting in the 2nd equation

    b+c = 10
(1+c)+c = 10
   1+2c = 10
     2c = 9
      c = 4.5   <-- but c² = 4.5² = 20.25 is not a square number.

Trying b-c = 2 and b+c = 5



Solve the first equation for b

b = 2+c

Substituting in the 2nd equation

    b+c = 10
(2+c)+c = 10
   2+2c = 10
     2c = 8
      c = 4 

Substitute c = 4 in 

    b+c = 10
    b+4 = 10
      b = 6

Substitute in

 a² + 5 =  b²
 a² + 5 =  6²
 a² + 5 = 36
     a² = 31

But 31 is not a square number.

So there is no solution.

Edwin