Factorial equations issues

vaungureanu · July 9, 2016, 11:42am

It’s my pleasure!

mhfard1370 · July 9, 2016, 2:11pm

please help me

mhelmle · July 9, 2016, 2:40pm

Regarding the original question of Valeriu Ungureanu I get the following results

In[1]:= eq1 = x! == ((7!)!)/(7!)

Out[1]= x! == \
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In[2]:= Solve[
 x! == 898662748881423000174453419447084561929984926188022983951261503\
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000000000000000000000000000 && x \[Element] Integers, {x}]

Out[2]= {{x -> 5039}}

The version of Mathematica I use is:

In[3]:= $Version

Out[3]= "10.4.1 for Microsoft Windows (64-bit) (April 11, 2016)"

vaungureanu · July 9, 2016, 2:50pm

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)"

vaungureanu · July 9, 2016, 3:40pm

Unfortunately, I don’t now the answer and I don’t want to make a supposition! Sorry, Mohamad!

vaungureanu · July 10, 2016, 4:50am

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.

mhfard1370 · July 10, 2016, 1:21pm

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:

? is injective
? is surjective
? is single-valued
so: ??¹ is single-valued.

Please let me know what you think.

vaungureanu · July 11, 2016, 7:09am

Dear Mohamad,

Did you use a Mathematica code to plot the graphs in the first comment? If so, can you expose that code?

mhfard1370 · July 11, 2016, 7:26am

sure

gianlucagorni · July 11, 2016, 8:29am

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

vaungureanu · July 11, 2016, 8:42am

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

vaungureanu · July 11, 2016, 8:45am

In[26]:= Solve[Gamma[x + 1] == (7!)!/7! && Abs[x - 5039] <= 10,
x] // N

During evaluation of In[26]:= Solve::nint: Warning: Solve used numeric integration to show that the solution set found is complete. >>

Out[26]= {{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}}

vaungureanu · July 11, 2016, 9:12am

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…

vaungureanu · July 11, 2016, 7:24pm

Solution plot…

t = x /. Solve[Gamma[x + 1] == (7!)!/7! && Abs[x - 5039] <= 10, x] // N // Quiet;
ListPlot[(Tooltip[{Re[#], Im[#]}] &) /@ t, AspectRatio -> 1]

gianlucagorni · July 11, 2016, 9:10pm

Could it possibly be periodic?

Plot[Log@Abs[Gamma[5039 + I y + 1] - (7!)!/7!], {y, -20, 20}, 
 PlotPoints -> 100]

vaungureanu · July 12, 2016, 5:18am

I made an experiment to find if there exists some periodicity… And found something very interesting… The real part of the solutions is increasing, e.g. {5050.53 - 999.855 I, 5050.53 + 999.855 I} are two conjugate solutions with real parts greater than initial integer solution 5039. More the more, we can observe that there exists something that looks appropriate to periodicity, but, unfortunately it is not or it is a very “special” periodicity. This may be observed in the results that I obtained after doing an experiment with the following code.

t = x /. Solve[Gamma[x + 1] == (7!)!/7! && Abs[x - 5039] <= 1000, x] // N
ListPlot[(Tooltip[{Re[#], Im[#]}] &) /@ t, AspectRatio -> 1]

As it took some time to obtain the results, I attached the nb file with the final solutions.

P.S. The second file contains results for Abs[x-5039] \leq 1100 with 2988 solutions…

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