[seqfan] Re: prime + factorial = prime

Jack Brennen jfb at brennen.net
Mon Mar 1 22:46:38 CET 2010

Leroy Quet wrote:
> But, maybe not. Has it been proved that there is always a factorial m! such that a prime p plus m! = a prime?
> As noted in the comment to sequence A092789, and which is obvious, m must be < p.

I did some quick searching, looking for probable primes (not doing
primality proofs)...

Only a very few primes p even require searching for m past sqrt(p),
and they are:

p=3, m=2
p=7, m=3
p=1609, m=42
p=74887, m=276

I think that although a proof might be difficult to impossible,
the heuristics will strongly suggest that there always exists a
solution m.  The number p+m! (when m<p) is so much more likely to
be prime than a random number of the same magnitude...

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