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# n!+1 is composite

## Question

How many n are there such that n! + 1 is composite?

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Since there are so many n's, I'm guessing the answer is Alepho

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Not familiar with the term in this usage -  What do you mean by composite??

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"Composite" refers to the number having at least two distinct positive integer divisors.

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"Composite" refers to the number having at least two distinct positive integer divisors.

Actually, that's not quite true as 1 is also a positive integer divisor.

Composite really just means "not prime".

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If p>3 is prime, consider the set of integers between 2 and p-2 inclusive. Each number in this set can be uniquely paired with another number in this set so that their product is congruent with 1 modulo p. Hence (p-2)! = 1 modulo p and consequently (p-1)! = -1 modulo p, so (p-1)!+1 is composite as it divides p. Let n=p-1 and n!+1 is composite, and since there are infinitely many primes there are infinitely many such n.

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"Composite" refers to the number having at least two distinct positive integer divisors.

Actually, that's not quite true as 1 is also a positive integer divisor.

Composite really just means "not prime".

Thanks

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Oops, my mistake

Edited by plasmid
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"Composite" refers to the number having at least two distinct positive integer divisors.

Actually, that's not quite true as 1 is also a positive integer divisor.

Composite really just means "not prime".

Not quite true.  "The number one is a unit; it is neither prime nor composite."  Prime and composite are terms that are applied to positive numbers (though the definitions can been extended to include negative numbers -- as associates to the positive, i.e., the negative number is deemed the same prime or composite as the positive number by the extended definition [refer to the Prime Pages FAQ "Can negative numbers be prime?"]). A composite number has been defined as any positive integer that has at least two prime factors.

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n! + 1 is prime for n = {0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209}
Other factorial primes n! + 1 may exist, yet none have been found (as prior to May 2014). From the generally accepted mathematical definition of composite, most other values of the natural numbers n! + 1 are composite. That is, within the domain of natural numbers, bonanova's guess, Aleph-0, is likely correct.
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Edited by DejMar

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