Is 1! + 11! + 300! prime?

[Jonathan Post]

I agree, Hugo.  I was thinking of 4 sequences:
(1) The array of n and k such that n! + k! + 1 = a prime, n<k, read by
antidiagonals;
(2) a(n) = the least k such that n! + k! + 1 = a prime, or 0 if none;
(3) a(n) = the greatest k such that n! + k! + 1 = a prime, or 0 if none;
(4) a(n) the number of solutions, i.e. the number of k such that  n! + k! +
1 = a prime, from 0 to upper bound A051301(n).

Or is there a way to compress this into fewer sequences?

I'm not saying that it's inherently interesting, but that it builds on
existing OEIS seqs such as A002981, A088332, A089359, is elementary, is new,
is not trivial, allows some number theory to be taught to non-science-major
Math students such as I had for many semesters.

I know quite a few 1 + A! + B! + C! primes but those don't compress into a
few sequences, 3-D arrays not as easy as 2-D.



On 8/14/06, Hugo Pfoertner <all at abouthugo.de> wrote:
>
> Joshua Zucker wrote:
>
> [...]
>
> > So someone
> > should still check 693 and 845 ...
>
> PFGW Version 1.2.0 for Windows [FFT v23.8]
> Primality testing 693!+11!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 13
> 693!+11!+1 is PRP! (69.7657s+0.0193s)
> Done.
>
> PFGW Version 1.2.0 for Windows [FFT v23.8]
> Primality testing 845!+11!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 13
> Running N-1 test using base 29
> 845!+11!+1 is PRP! (237.4787s+0.0485s)
> Done.
>
> The given times are on a 800 Mhz Athlon, running another job. I am
> currently preparing tests for the next round of Al Zimmermann's
> programming contest, which will be on self-avoiding zigzag paths
> ("matchstick snakes")
>
> >
> [...]
> >
> > For 1! + 17! + n!, my system produced the following candidate primes:
> > 11 14 46 183 560
>
> Confirmed with pfgw:
> 11!+1+17! is 3-PRP! (0.1049s+0.9518s)
> 14!+1+17! is 3-PRP! (0.0001s+0.0538s)
> 46!+1+17! is 3-PRP! (0.0003s+0.0415s)
> Switching to Exponentiating using Woltman FFT's
> 183!+1+17! is 3-PRP! (0.9605s+0.9951s)
> 560!+1+17! is 3-PRP! (30.1122s+6.2870s)
>
> checked:
> PFGW Version 1.2.0 for Windows [FFT v23.8]
> Primality testing 560!+17!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 19
> Running N-1 test using base 23
> 560!+17!+1 is PRP! (75.0765s+0.0597s)
> Done.
> Primality testing 183!+17!+1 [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 19
> Running N-1 test using base 23
> Running N-1 test using base 37
> 183!+17!+1 is PRP! (9.1043s+0.0516s)
> Done.
> >
> [...]
> > --Joshua Zucker
>
> To JVP:
> Please don't conclude from my contribution that I find this problem very
> interesting; I just wanted to confirm Joshua's results.
>
> What I definitely don't support is a series of new sequences:
>
> Numbers n such that n!+k!+1 is prime, for k=1...100.
>
> Hugo Pfoertner
>
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