Question
If x,y are positive integers such that 2^x+49 = y^2,
find the value of x and y.
If x,y are positive integers such that 2^x+49 = y^2,
find the value of x and y.
Since 2^x are multiples of 2, 2^x is even. Addition of even and odd numbers results in an odd number. Therefore, y^2 is odd , so it is for y
So y-7 and y+7 are numbers of power of 2
Let
Then
2y = 2^k+2^m =2^k(1+2^{m-k})
y = 2^{k-1}(1+2^{m-k})
To make y be odd, the only possible is to make
k-1 = 0, that is
k =1
Substitutng to (3) gives
y = 7+2 = 9
Substitute y to (2)
2^x = (9-7)(9+7) = 32
x = 5