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Factorial equations issues

A very simple equation: $x!=\frac{(7!)!}{7!}$

It's easy to find the solution:

x = 7! - 1 = 5039

I used Mathematica to solve this equation:

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

No one of these functions gives the solution!

Can someone comment this strange?

35 Replies

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... enter image description here

Attachments:

Could it possibly be periodic?

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

Solution plot...

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

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

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
POSTED BY: Gianluca Gorni

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)"
Posted 9 years ago

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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9304733848531425261563826553653736458389370786339309690991932519148709\
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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)"
POSTED BY: Michael Helmle
Posted 9 years ago

please help me

Attachments:
POSTED BY: Mohamad Fard

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

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.

Posted 9 years ago

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.

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POSTED BY: Mohamad Fard

Dear Mohamad,

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

Posted 9 years ago

sure

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POSTED BY: Mohamad Fard

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...

Posted 9 years ago

thanks a lot.

POSTED BY: Mohamad Fard

It's my pleasure!

Posted 9 years ago

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?

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POSTED BY: Mohamad Fard

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!

I don't know surely how the Mathematica system solves this equation. I can suppose only that the Mathematica system uses methods of discrete mathematics and it obtains the result by applying its principles of expression representation in the "simplest" form. So, in this case we can't discuss about an inverse function of factorial. Once again, it's only my opinion!

Posted 9 years ago

thanks.

POSTED BY: Mohamad Fard
Posted 9 years ago

for x Real : x?1

a = x! => x = ?

POSTED BY: Mohamad Fard
Posted 9 years ago

Thank you, too. if you know which mathematical formula is used to solve x!=5039! please tell me. Regards

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POSTED BY: Mohamad Fard

Thank you, Louis!

For me is somewhat strange that if we do not specify that x is integer, then we do not obtain the result...

The Set of Reals doesn't contain the Set of Integers?

What is the principle that governs the execution of Solve function?

Posted 9 years ago

Try this:

Solve[(7!)!/(7!) == x!, x, Integers]
POSTED BY: Louis Godwin
Posted 9 years ago

could you please tell me exactly :

for x Real : x?1

  • x! = a => x = ?
POSTED BY: Mohamad Fard

Mohamad, may be the answer to you questions is in Gamma[] function as an extension of the factorial function.

In[29]:= Solve[Gamma[x] == 24, x]

During evaluation of In[29]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

Out[29]= {{x -> 5}}

In[34]:= Solve[Gamma[x] == 10, x]

During evaluation of In[34]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

Out[34]= {{x -> 
   Root[{-10 + Gamma[#1] &, 4.39007765083314189217115670719}]}}

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?

Posted 9 years ago

You're welcome. well, I think the Problem isn't simple. we only write: solve(x!=a) but the problem is: there is not an standard mathematical formula to inverse of factorial function, you know. for example when we write: x!=a=>x=? and when we use (=) , we only want exact solution of problem but which function can calculate inverse of factorial? none. so we can only use special functions for special conditions or approximate solutions.

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POSTED BY: Mohamad Fard

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?

Posted 9 years ago

Or this formula:

floor(3/4+e^(1+ProductLog(-(1+log((2 ?)/x^2))/(2 e))))

POSTED BY: Mohamad Fard
Posted 9 years ago

u can use this formula:

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POSTED BY: Mohamad Fard

???

Posted 9 years ago
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