Wikipedia:Reference desk/Archives/Mathematics/2017 August 3

= August 3 =

Shannon coding and graphs in a video
A little more than an hour into this video, some graphs are shown that Gates says are related to Shannon coding. How do they come about from Shannon coding? Bubba73 You talkin' to me? 00:56, 3 August 2017 (UTC)
 * I'm guessing what they're actually referring to are Signal-flow graphs. --RDBury (talk) 17:21, 4 August 2017 (UTC)

Primality of least positive m such that n!+m is prime
This is A033932, which conjectures that no term is a composite number. Would this conjecture follow from any other well-known conjecture on the primes? Or why should we expect it to be true, beyond bruteforce numerical evidence? 68.0.147.114 (talk) 22:30, 3 August 2017 (UTC)
 * Note that m = 1 for factorial primes n!+1, and 1 is neither prime nor composite. But large factorial primes are rare.
 * If m has a prime factor p ≤ n then p divides n! = 1×2×...×p×...×n. Then p also divides n!+m. So if n!+m is prime then m has no prime factor ≤ n. If m is composite then it must be the product of at least two primes above n, so m > n2. If m is the least positive m such that n!+m is prime then it would mean there are no primes from n! to n! + n2. That seems unlikely based on the typical size of prime gaps. See also the related fortunate numbers. PrimeHunter (talk) 23:12, 3 August 2017 (UTC)


 * It is known as "Fortune's Conjecture" and I think it was in Scientific American mathematical games column about December 1980. It would not follow from other conjectures as far as I know.  Such numbers are more likely to be prime because they can't be a multiple of a prime <= n.  Probably if you don't find a small counterexample, the changes of finding one get smaller and smaller.  Bubba73 You talkin' to me? 01:54, 4 August 2017 (UTC)


 * Aside: (Suggested improvement to related article.) Factorial prime lists the 49 factorial primes known as of December 2106.  Can we get a reference to the conjecture that there are infinitely many factorial primes and their conjectured density?  A referenced mention of projects searching for additional factorial primes would also be welcome.  Cheers! -- ToE 15:30, 6 August 2017 (UTC)