Question

If x,y are positive integers such that 2^x+49 = y^2,

find the value of x and y.

Collected in the board: Number Theory

Steven Zheng posted 15 hours ago

Answer

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

2^x+49 = y^2
(1)
2^x = y^2-49=(y-7)(y+7)
(2)

So y-7 and y+7 are numbers of power of 2

Let

y-7 = 2^k
(3)
y+7 = 2^m
(4)

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

Steven Zheng posted 15 hours ago

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