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bradley_lee_r
Dec 31, 2019

Rational solutions to the equation y^x=x^y

I run a Facebook Group called not just tiny-c Programming Group. A member recently posted the above problem. This is actually a very old problem but we were encouraged to give it a shot without looking it up. The original problem was this: graph the above equation.


For "extra credit" we were encouraged to solve it for rational numbers. An obvious solution is x=y. Also, 2^4=4^2. It turns out there are infinitely many rational numbers that solve this equation.


I wrote a computer program in C which approximately solved our equation. I populated a table with the x,y values it generated into desmos and got a pretty good graph. Actually, you can just put the equation itself into desmos and get the exact graph.


But the rationals was too tough for me. I eventually did the research and got the answer. Very, very interesting.


I watched blackpenredpen's videos on this. I doubt this rational problem would have enough general interest to make a video on this but ... you never know.



2 comments
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bradley_lee_r
Dec 31, 2019

The "approximate" way to get the real values is done by the C program at


http://primepuzzle.com/mouse/xyyx.c


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Faizan Ahmed
Mar 10, 2020

When x and y both are equal and non

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