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Re: Are some equations unsolvable?

This is what I have:

f = (1/(sigma*Sqrt[2 Pi]))*Sqrt[(Pi/(-1/(2*sigma^2)))]*
  Exp[((m/sigma^2)^2)/4*(-1/2*sigma^2) + (m^2/(2*sigma^2))]

Solve[f == 216, sigma]

And I get this message: "This system cannot be solved with the methods available to Solve" 

Is it because there is no way to isolate sigma? Or am I doing something wrong?


Some are unsolvable indeed. And not only due to limitations of Mathematica, but just due to their internal nature.
In addition come limitations are imposed also by Mathematica. If you look into Menu/Help/Solve/More Information you find the following:

Solve deals primarily with linear and polynomial equations. 
The option InverseFunctions specifies whether Solve should use inverse functions to try and find solutions to more general equations. The default is InverseFunctions->Automatic. In this case, Solve can use inverse functions, but prints a warning message. See notes on InverseFunctions. 

That is, some non-polynomial equation it can solve, but not all of them.
It is a good idea to help Mathematica understand, of what do you need to achieve. In your case, for example, you may have specified, if the variables are real or complex. However, 
This is your left hand part:

In[19]:= Clear[f];
f = (1/(sigma*Sqrt[2 Pi]))*Sqrt[(Pi/(-1/(2*sigma^2)))]*
  Exp[((m/sigma^2)^2)/4*(-1/2*sigma^2) + (m^2/(2*sigma^2))]

ff = Simplify[f /. sigma -> x*m, {m > 0}]

Out[20]= (\[ExponentialE]^((3 m^2)/(8 sigma^2)) Sqrt[-sigma^2])/sigma

Out[21]= (\[ExponentialE]^(3/(8 x^2)) Sqrt[-x^2])/x

Now it is clear that if solution exists, it is imaginary:

In[17]:= FindRoot[(\[ExponentialE]^(3/(8 x^2)) Sqrt[-x^2])/x, {x, 

Out[17]= {x -> 0.+ 0.0592883 \[ImaginaryI]}

Have fun, Alexei

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