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2 solutions
Moderator note:
This solution doesn't answer the problem.
While this technically would work in most cases because the factorial's value is greater than the sum, your proof has flaws. Firstly, I think you misinterpreted the factorial notation.
Second thing to note is that the question asks if 1 9 9 2 ! − 1 is a prime or not but your proof answers if 1 9 9 2 ! is a prime or not.
Using Wilson Theorem,
1 9 9 2 ! ≡ 1 m o d 1 9 9 3
1 9 9 2 ! − 1 ≡ 0 m o d 1 9 9 3
so it's not a prime...
Moderator note:
As pointed out by Brian, this solution is completely wrong.
Wouldn't Wilson's Theorem assert that, since 1 9 9 3 is prime,
1 9 9 2 ! ≡ − 1 ( m o d 1 9 9 3 ) ⟹ 1 9 9 2 ! + 1 ≡ 0 ( m o d 1 9 9 3 ) ?
WolframAlpha does confirm that 1 9 9 2 ! − 1 is indeed not prime, but as far as I can tell no analytic proof of this fact has been established. This sequence contains all n such that n ! − 1 is prime, but it is obtained through computational and not analytic methods.
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It appears that 1 9 9 2 ! − 1 is divisible by 3449 and also by 8627.
By the way, Brian, is there such a thing as an "analytic proof" that a number is prime? Isn't it all just (more or less clever) computation?
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Haha. Good point; it's more about the most efficient algorithm you can create. To that end, I just noticed that this was a computer science question and not number theory, so I suppose rohit was looking for just such an algorithm.
any proof??
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@Dev Sharma – Like most issues surrounding primality, this is done by computation.
Wolfram Alpha returns that 1 9 9 2 ! − 1 is not a prime in my case
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Yes, you're right, I just forgot to type "not" in front of "prime". :P
how to check on wolfram alpha?
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You can either enter "is 1992! - 1 prime", the response just being that it is not a prime, or "factor 1992! - 1", which yields that 3449 and 8627 are prime factors, (although a full factorization is not provided).
Well I have never read wilson theorem. I just made an analysis, based on that my answer should be prime. Well does this quest. was specifically for computer science, cause I don't go to school. I am illerate 2!-1= 1 3!-1 = 5 4!-1 = 23 5!-1 = 119 6!-1 = 719 . . 10!-1 = 3628799 . . 1992!-1 = prime.
1+9+9+2 = 21 which is a multiple of 3 so 3 devides 1992 so it's not a prime