Find all pairs of integers x and y such that x^2-y^2=104
(x + y)(x - y) = 104
Now factor 104:
are all the possibilities.
Now you have to solve simultaneous equations:
x + y = 1
x - y = 104
The solution to this is x = 105/2 and y = -103/2, which aren't integers so this doesn't work. So move on to the next possibility. etc.
I note that there are solutions for all pairs of factors where both factors are even.
x + y = a
x - y = b
where a*b = 104
Solve the top equation for y:
y = a - x
Then insert this value of y into the bottom equation:
x - (a - x) = b
2x - a = b
x = (b + a)/2
y = a - x = a - (a + b)/2 = (b - a)/2
In order for x and y to be integers we need a + b and a - b to be even. Thus a and b must either both be odd or must both be even. Since a*b = 104 (an even number) at least one of a and b must be even.
Thus both a and b must be even numbers for x and y to be integers.