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 May 23rd, 2013, 10:21 PM #1 Senior Member   Joined: Apr 2013 Posts: 425 Thanks: 24 An equation with the factorial Hello! To solve the equation $x!+2(x-1)!+1-y^2=0$ where $x,y\in Z$. Thank You!
 May 24th, 2013, 02:07 AM #2 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: An equation with the factorial Interesting question, I found (1, 2), (5, 13) and (6, 31) using brute-force upto 10^4 but found no other. I would like to see a proper diophantine approach to this one. PS : I think this should belong to number theory section.
May 24th, 2013, 02:29 AM   #3
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Re: An equation with the factorial

Quote:
 Originally Posted by mathbalarka Interesting question, I found (1, 2), (5, 13) and (6, 31) using brute-force upto 10^4 but found no other. I would like to see a proper diophantine approach to this one.
How were you able to go to 10^4?
1000! has 2568 digits

How many does (10^4)! have?

May 24th, 2013, 02:36 AM   #4
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Re: An equation with the factorial

Quote:
 Originally Posted by Denis How were you able to go to 10^4?
It's a matter of fact that PARI handle such large brute-force. I think I can push it to c*10^5 for some c > 0 since I have developed the complexity of my code.

Quote:
 Originally Posted by Denis How many does (10^4)! have?
35660.

May 24th, 2013, 02:44 AM   #5
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Re: An equation with the factorial

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by Denis How were you able to go to 10^4?
It's a matter of fact that PARI handle such large brute-force. I think I can push it to c*10^5 for some c > 0 since I have developed the complexity of my code.
Ah, I see....

Btw, I don't see that problem any different from (as example):
F(n)x^2 + 2(F(n-1))x + 1 - y^2 = 0 ; F = Fibonacci sequence

I mean "generally" same...

 May 25th, 2013, 05:54 AM #6 Senior Member   Joined: Apr 2013 Posts: 425 Thanks: 24 Re: An equation with the factorial Hello! An idea: 1.-The equation can be written $(x-1)!(x+2)=y^2-1$ and so it follows that $y=\pm (2u+1)$ for $x>1$ where $u\in N*$.What is the connection between the sum of the first natural numbers and the number $y$? 2.-The equation can be written $(x+2)!=(y^2-1)x(x+1)$. 3.-How many prime numbers $y$ satisfy the equation $x!+2(x-1)!+1-y^2=0$? Other ideas .... Thank you very much!

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