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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 where . 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 bruteforce 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  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: An equation with the factorial Quote:
1000! has 2568 digits How many does (10^4)! have?  
May 24th, 2013, 02:36 AM  #4  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: An equation with the factorial Quote:
Quote:
 
May 24th, 2013, 02:44 AM  #5  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: An equation with the factorial Quote:
Btw, I don't see that problem any different from (as example): F(n)x^2 + 2(F(n1))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 and so it follows that for where .What is the connection between the sum of the first natural numbers and the number ? 2.The equation can be written . 3.How many prime numbers satisfy the equation ? Other ideas .... Thank you very much! 

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