1961 AMC 12 Problem 33

Question

The number of solutions of

2^{2x}-3^{2y}=55,

in which

x

and

y

are integers, is:

0

1

2

3

more than three, but finite

Factor the left side as a difference of squares

The two positive factors multiply to

55

and have the same parity

Factor:

(2^x-3^y)(2^x+3^y)=55.

First,

x>0.

If

y\le0,

then

55<4^x\le56,

which is impossible for an integral power of

4.

Hence

y>0,

so both factors are positive odd integers. The factor pair

5,

11

gives

2^{x+1}=16

and

2\cdot3^y=6,

so

(x,y)=(3,1).

The factor pair

1,

55

would require

2^x=28,

impossible. Hence there is exactly one solution.

Thus, the correct answer is B.

Problem 33 in other years