Is 1! + 11! + 300! prime?

[Joshua Zucker]

On 8/13/06, Jonathan Post <jvospost3 at gmail.com> wrote:
> 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.

Just to clarify: I am running PLT DrScheme, not PARI, and doing
Miller-Rabin tests, so I am not factorizing the numbers but just
proving (most of) them composite.  The values of n I list (up to 1000)
are only candidate primes, not proven primes by my method.  So someone
should still check 693 and 845 ...

It took my machine a couple hours to check up to 1000, with the time
growing as the numbers grew (more than 20 minutes to get from 900 to
1000, compared with 15 minutes to get from 700 to 800), so getting up
to 11! would be impractical with my methods.

> 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?
>

For 1! + 17! + n!, my system produced the following candidate primes:
11 14 46 183 560

Now, we agree on 183, which is good.
And it may well be that some of mine here are not really primes --
like I say, they're only probable primes.  So someone still has to
check 560!

But the ones I reject really should be composite, so I am very
suspicious of your 42, and it shouldn't be necessary to check anything
other than the numbers given here.  In fact I believe 1! + 17! + 42!
is divisible by 199.

I suppose it makes sense not to repeat the 11 or the 14, since they
were counted already.  Probably your 42 was just a typo and should
have been 46?

--Joshua Zucker