Prime Sums of Exactly Three Distinct Factorials

[Jonathan Post]

It is the general case, for fixed A, distinct from B, that A! + B! + 1! =
prime has no solutions, or several solutions with an upper bound of
A051301(A) = the least prime factor of A! + 1.

The number of such solutions, for A = 1, 2, 3, ... is
(0, 0, 3, 0, 0, 0, 9, 4, 3, 0, ...) where I don't yet know the 11th term,
since 11! + 1 = 39916801 is prime.  There is, by search and Alpertron, no 85
< B <= 270 such that 11! + 1! + B! is prime, but it is not easy to find if
42, 77, 85, are all the solutions, or if there are many more beyond my
current search.

Thanks for your correct comment, which earns you an acknowledgment in the
short paper that should come from this: elementary, not trivial, nicely
connected to OEIS, maybe for "Mathematics Magazine" or other venue for
college math teachers and clever high school students.

On 8/11/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> For *any* k > 1, the number of primes of the form n!+k is finite.  In
> particular, n!+k is divisible by k whenever n>=k.  More generally, when
> n >= any divisor of k > 1.
>
> In this case, k = A! + 1.
>
> In particular, 7!+1 = 5041 = 71^2, so 7! + n! + 1! cannot be prime for
> n >= 71.  Meaning Jonathan has probably found all of them.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Jonathan Post <jvospost3 at gmail.com>
> ...
>   It is the case in general, for fixed A distinct from B, that A! + B! +
> 1! = prime has either no solutions, or several solutions after which
> all higher A gives the same common prime factor to A! + B! + 1!, and it
> is not clear under what circumstances "several" means infinite.
> ...
> 7! + n! + 1! is prime for n = {8, 12, 16, 23, 27, 33, 37, 42, 53, ?}.
>
> ...
>
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