Is 1! + 11! + 300! prime?

[Jonathan Post]

Thank you all!  The difference in speed, at this scale, between PARI (on
Joshua Zucker's system) and the Alpertron running on my Linux box is
dramatic.  The factorization of 1! + 11! + 300! took 6 hours + 46 minutes
for me. For 1! + 11! + B! to be prime, the upper bound for B is the prime
11! +1, which leaves us a LOT of room for more solutions.

There are no solutions to 1! + 12! + B! = prime.
The complete solutions to 1! + 13! + B! = prime are 21, 26, 29, 44, 45.
The complete solutions to 1! + 14! + B! = prime are 16, 17, 18, 22.
The complete solutions to 1! + 15! + B! = prime are 19, 20, 21, 29.
There are no solutions to 1! + 16! + B! = prime.

So the next open case is  1! + 17! + B! = prime, where the upper bound of B
is least prime factor of 17! + 1, namely 661.  The two smallest solutions to
1! + 17! + B! = prime are B = 42, B = 183. 1! + 17! + 183! is a 337-digit
prime that my system verified in 20 minutes.  What are the other solutions
to 1! + 17! + B! = prime, for 183 < B < 661?

There are no solutions to 1! + 18! + B! = prime.
The complete solutions to 1! + 19! + B! = prime are 23, 26, 38, 42, 45, 50.

On 8/13/06, Joshua Zucker <joshua.zucker at gmail.com> wrote:
>
> I confirm Joseph's result: 1! + 11! + 300! is indeed divisible by 2777.
> Running some primality tests (of the sort that can prove a number
> composite but don't necessarily prove a number prime) I find up to n =
> 1000 that the only candidate primes are when n = 17 18 21 42 77 85 693
> 845
>
> Anyone want to look more carefully into 1! + 11! + 693! or + 845! perhaps?
>
> But trying it up to 11! seems like it's going to be way too
> time-intensive...
>
> --Joshua Zucker
>
> On 8/12/06, Joseph Biberstine <jrbibers at indiana.edu> wrote:
> > AFAICT 2777 is a factor.
> >
> > Jonathan Post wrote:
> > > Is 1! + 11! + 300! prime? It has 602 digits, and the Alpertron calls
> it
> > > "unknown" as to primality after half an hour on my Linux box. This is
> > > the first such case I've encountered in my A! + B! + 1! primality
> search.
> >
> >
>
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