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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)[reply]

I'm guessing what they're actually referring to are Signal-flow graphs. --RDBury (talk) 17:21, 4 August 2017 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]