Try this:

Solve[(7!)!/(7!) == x!, x, Integers]

In the complex domain there seem to be other solutions near the integer solution 5039:

Reduce[Gamma[1 + x] == (7!)!/7! && Abs[5039 - x] <= 2, x] // N

Dear Valeriu

Thank you for your time. well, I think ??¹ is Multivalued function without restriction of ?.

but I restrict ? to => x Real : x?c.

then:

1. ? is injective
2. ? is surjective
3. ? is single-valued
4. so: ??¹ is single-valued.

Please let me know what you think.

Attachments:

I humbly thank you but how do the computer calculate integral or series to infinity?

are the calculation of mathematica using integral or taylor series approximate formulas?

Attachments:

Unfortunately, this is not a code! I obtained the same result by applying a plain English query: = inverse(gamma(x))

It's interesting that an appropriate query: =Inverse[Gamma[x]], gives other result without plotting the graph of $\Gamma^{-1}$ function.

I wanted to plot the graph of $\Gamma^{-1}$ function to show that the issue related to multi-value function remains in the case when $x\geq c \geq 0$.

What I highlighted in this discussion is that to construct a graph of a inverse function is very easy... We must only change the axis ("transpose" the graph), what I wanted to do. But to construct the inverse of a function is a difficult problem...

So, the problem you posed remains, but the situation is much more clear...

Marvelous!

In[3]:= Reduce[Gamma[1 + x] == (7!)!/7! && Abs[5039 - x] <= 10, x] // N

During evaluation of In[3]:= Reduce::nint: Warning: Reduce used numeric integration to show that the solution set found is complete. >>

Out[3]= x == 5039. || x == 5039. - 0.737025 I || 
 x == 5039. + 0.737025 I || x == 5039. - 1.47405 I || 
 x == 5039. + 1.47405 I || x == 5039. - 2.21108 I || 
 x == 5039. + 2.21108 I || x == 5039. - 2.9481 I || 
 x == 5039. + 2.9481 I || x == 5039. - 3.68513 I || 
 x == 5039. + 3.68513 I || x == 5039. - 4.42215 I || 
 x == 5039. + 4.42215 I || x == 5039. - 5.15918 I || 
 x == 5039. + 5.15918 I || x == 5039. - 5.8962 I || 
 x == 5039. + 5.8962 I || x == 5039. - 6.63323 I || 
 x == 5039. + 6.63323 I || x == 5039. - 7.37025 I || 
 x == 5039. + 7.37025 I || x == 5039. - 8.10728 I || 
 x == 5039. + 8.10728 I || x == 5039. - 8.8443 I || 
 x == 5039. + 8.8443 I || x == 5039. - 9.58133 I || 
 x == 5039. + 9.58133 I

Dear, Mohamad,

I can suggest you an answer to your question! Observe the graph of the function $\Gamma^{-1}$. It's enough to look at the picture you posted in the first comment of this discussion. The function $\Gamma^{-1}$ is a multi-valued (set-valued) function. So, the mathematical domain/area of such functions is larger than the traditional mono-valued calculus and as it seems you can't have a simple representation (formula) of the frunction $\Gamma^{-1}$ as you want.

Thank you, Michael! It works fine if we state that x is integer.

In[8]:= eq = x! == (7!)!/7!;
Solve[eq && x \[Element] Integers, x]
Solve[eq, x, Integers]
$Version

Out[9]= {{x -> 5039}}

Out[10]= {{x -> 5039}}

Out[11]= "10.4.1 for Microsoft Windows (64-bit) (April 11, 2016)"

It's my pleasure!

Sure Mathematica can calculate Improper Integrals!

Mathematica is a computer algebra system, i.e., it was, from its creation, designed by Stephen Wolfram to deal with symbolic calculus.

You are welcome!

thanks.

for x Real : x?1

a = x! => x = ?

It's important to observe that Mathematica can treat such type of equation, e.g.,

Solve[x! == 7!/7, x]

It gives the correct solution x = 6.

As I understand the equation

x! == (7!)!/7!

may be reduced to equation

 x! == 5039!

by simplifications, and then it may give the correct result.

Why Mathematica doesn't make such transformations?

Dear Mohamad,

Thank you for your comments!

But my question is more deep... Mathematica is aimed for different users, including children. Can you imagine their reactions when such a powerful system as Mathematica can't solve a very simple problem?

What are the real issues which don't permit Mathematica to solve such problems?

u can use this formula:

Attachments:

http://community.wolfram.com/groups/-/m/t/882519?p_p_auth=XDmOe39D