Is 1! + 11! + 300! prime?

[Gerald McGarvey]

PARI shows 1! + 17! + 42! = 199 * 2049293 * 21763438520903 * 
158304612615480381626247252181
but that 1! + 17! + 46! is prime
and that 1! + 17! + n! is prime for n = 11, 14, 46, 183, and 560

-- Gerald

At 01:47 PM 8/13/2006, Joshua Zucker wrote:
>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