BMAD Posted March 12, 2015 Report Share Posted March 12, 2015 How many n are there such that n! + 1 is composite? Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted March 12, 2015 Report Share Posted March 12, 2015 Since there are so many n's, I'm guessing the answer is Aleph_{o} Quote Link to comment Share on other sites More sharing options...

0 dgreening Posted March 12, 2015 Report Share Posted March 12, 2015 Not familiar with the term in this usage - What do you mean by composite?? Quote Link to comment Share on other sites More sharing options...

0 Perhaps check it again Posted March 12, 2015 Report Share Posted March 12, 2015 "Composite" refers to the number having at least two distinct positive integer divisors. Quote Link to comment Share on other sites More sharing options...

0 gavinksong Posted March 13, 2015 Report Share Posted March 13, 2015 "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". 1 Quote Link to comment Share on other sites More sharing options...

0 Rainman Posted March 13, 2015 Report Share Posted March 13, 2015 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. 1 Quote Link to comment Share on other sites More sharing options...

0 dgreening Posted March 13, 2015 Report Share Posted March 13, 2015 "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 Quote Link to comment Share on other sites More sharing options...

0 plasmid Posted March 13, 2015 Report Share Posted March 13, 2015 (edited) Oops, my mistake Edited March 13, 2015 by plasmid Quote Link to comment Share on other sites More sharing options...

0 DejMar Posted March 13, 2015 Report Share Posted March 13, 2015 "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. Quote Link to comment Share on other sites More sharing options...

0 DejMar Posted March 13, 2015 Report Share Posted March 13, 2015 (edited) 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.. Edited March 13, 2015 by DejMar Quote Link to comment Share on other sites More sharing options...

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## BMAD

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

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