Solutions to the equation yny=1 for even n
Solutions to the equation for even n
Answer & Explanation
A long time ago I was curious about the closed-form solutions to the equation:
For n an odd number, try . Then . This gives the formula
which can only be true if and n is odd (k cannot be an integer for the formula to work, check this youself.)
Furthermore one has to have that
which is real if n is odd.
Thus there are closed-form solutions to my problem for n odd, and my question is if anyone can find a closed-form solution for or in general if n is even.
Well i tried for , and i have partial solution. I will use this reply to trace the work since it supports LaTeX, I will try to complete it later (or someone else can take over).Suppose Then ,or .So where .Solving , we haveNow we need to solve for f(y), and invert the function to get