Ankit Khokhar
Last Activity: 11 Years ago
Ok lets see, write the equation as:
x^2 = y^2 + 2
Now, so
x = (y^2 + 2)^(1/2)
Now the above equation represents a hyperbola ( by general form: (x^2)/(a^2) - (y^2)/(b^2) =1 )
so, it will be symmetric,
so if it have any integral coordiantes ( or any integral solution of above equation in any of the four quadrants then it should have in all four , and if not in any of one then in none of them )
so lets take out the first quadrant,
here x and y, both are positive,
Now vary the value of y so that we can get corresponding values of x (and can see if both be integers)
lets start by least integer (considering the first quadrant) which is zero, 0
so if y = 0,
x=(2)^(1/2)=1.414 , not an integer
now let y = 1, so x=(3)^(1/2) = 1.732,not an integer again
now let y=2, so x=(4 + 2)^(1/2) = 6^(1/2), again not an integer
Now from 2 and above, the perfect squares will be at a distance more than 2, so cannot get a perfect square by adding a perfect square with 2
REASON: we have to get: Perfect square = another Perfect square + 2
Now for 2, square = 2*2
for 3 square = 3*3,
so the distance between consecutive square increases as the number increases,
so no integral solutions can be obtained afterwards,
Best of Luck for your exams