Solve this equation with p&p to find two nice solutions.
72^{x}ยท48^{y}=6^{xy}
The person who shared this problem on Twitter and posted a video of their solution on YouTube just found these two solutions. A bunch of people who replied on Twitter just found these two solutions.
However, if you use your favorite graphing calculator, you will see there are many more solutions.
Find a set of parametric equations that will generate all solutions.
I don't know what p&p is, or why in the world this is rated as a hard problem, but here goes:
(2^3 * 3^2)^x * (2^4 * 3)^y = (2 * 3)^xy
One pair of solutions corresponding to matching powers of 2 and 3 on both sides. This requires
3x + 4y = 2x + y = xy
For which there are two solutions:
(x, y) = (0,0) or (5, 5/3)
But, more generally:
2^(3x + 4y  xy) = 3^(xy  2x  y)
(3x + 4y  xy) / (xy  2x  y) = log_2(3)
(3 + 2A) x + (4 + A) y = (A +1) xy where A = log_2(3) has a continuum of solutions

Posted by FrankM
on 20230623 20:44:00 