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May 23rd, 2013, 10:21 PM   #1
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An equation with the factorial

Hello!
To solve the equation where .
Thank You!
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May 24th, 2013, 02:07 AM   #2
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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.
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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?
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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.
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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...
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May 25th, 2013, 05:54 AM   #6
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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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