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Number Theory Level 4

Let $(x_1, y_1), (x_2, y_2) , \ldots , (x_n , y_n)$ be all the solutions of integers $(x,y)$ that satisfy the equation

$\large 3^x 4^y = 2^{x+y} + 2^{2(x+y) - 1}$

What is $\displaystyle \sum_{j=1}^n (x_j + y_j)$ ?

12 10 7 6 8

1 solution

Hana Wehbi
May 9, 2017

The right side is $2^{x+y}(1+ 2^{(x+y)-1})$ . If the second factor is odd, it needs to be a power of $3$ , so the only options are $x+y = 2$ and $x+y = 4.$

This leads to two solutions, namely (1,1) and (2,2).

The second factor can also be even, if $x + y - 1 = 0.$

Then $x+y=1$ and $3^x4^y= 2+2$ giving $(0,1)$ as the only solution.

Thus $1+1+2+2+1+0=7 .$

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