[seqfan] Re: prime + factorial = prime

T. D. Noe noe at sspectra.com
Mon Mar 1 22:56:17 CET 2010

```At 1:46 PM -0800 3/1/10, Jack Brennen wrote:
>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...

For each prime, you should be able to compute the number of m for which
p+m! is prime. For the first 50 primes, I obtain

1, 1, 3, 4, 5, 3, 6, 7, 6, 6, 9, 11, 9, 5, 10, 9, 10, 9, 9, 8, 9, 9, 11, 8,
10, 10, 12, 16, 12, 10, 10, 13, 14, 14, 16, 11, 12, 9, 15, 10, 9, 8, 12, 9,
10, 6, 8, 7, 14, 13

Quite a few!

Tony

```