Proof

[Richard Mathar]

>  If in title you will be change on
>  a(n) = smallest prime p such that (n+p!)/n is prime
>  and all will be as good as this is possible in this case
>
>  without  "or 0 if no such prime exists."
>
>  because we can't put zero without proof which can't exist recently that
> after deleting "or 0 if no such prime exists."

But it is well possible that there is a proof for certain n that
1+p!/n cannot be a prime, e.g. like in Sierpinski Riesel case, one
might detect a periodicity in the sequence 1+k!/n (and prove it)
showing that it may only be prime for certain k, and then show that no
such k may ever be prime.

But the fundamental difference is :
with "0 if no such p exists" the sequence IS well defined (whether or
not we will ever be able to know a(9))
while without this phrase, the sequence is NOT well defined, unless
you prove that such a prime will always exist.

Maximilian




Removing typos from my previous e-mail we have...

aj> From seqfan-owner at ext.jussieu.fr  Wed Apr 23 17:47:33 2008
aj> Date: Wed, 23 Apr 2008 17:49:24 +0200
aj> From: Artur <grafix at csl.pl>
aj> Reply-To: grafix at csl.pl
aj> To: "N. J. A. Sloane" <njas at research.att.com>
aj> CC: seqfan <seqfan at ext.jussieu.fr>
aj> Subject: Proof
aj> ...
aj> I hope that I have proof:
aj> 1) 4 x^2 - 4 x*y + 7 y^2 can be odd prime only  if y is odd
aj> 4 x^2 - 4 x*(2 k + 1) + 7 (2 k + 1)^2=7 + 28 k + 28 k^2 - 4 x - 8 k x + 
aj> 4 x^2=
aj> 7+4(7 k + 7 k^2 - x - 2 k x + x^2)
aj> 2) Now if we solve equation (7 k + 7 k^2 - x - 2 k x + x^2) = m on 
aj> variable x and we will be forced x as positive integer
aj> we are receiving
aj> x=(1+2k+/-Sqrt[-24k^2-24k+4m+1])/2
aj> now to integer x condition need     -24k^2-24k+4m+1 have to be odd square
aj> but these have to be 24w+1 as Zak Seidov states in comment to A001318 
aj> <http://www.research.att.com/%7Enjas/sequences/A001318>
...

This is not clear to me: the -24k^2-24k+4m+1 can be odd squares without
being of the form 24w+1: they can (at least) also be of the form 24w+9 :

"k=", -4, "m=", 74, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", -4, "m=", 92, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8

"k=", -3, "m=", 38, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", -3, "m=", 56, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8

"k=", -3, "m=", 92, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8

"k=", -2, "m=", 14, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", -2, "m=", 32, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8

"k=", -2, "m=", 68, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8

"k=", -1, "m=", 2, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", -1, "m=", 20, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8

"k=", -1, "m=", 56, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8

"k=", 0, "m=", 2, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", 0, "m=", 20, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8

"k=", 0, "m=", 56, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8

"k=", 1, "m=", 14, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", 1, "m=", 32, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8

"k=", 1, "m=", 68, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8

"k=", 2, "m=", 38, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", 2, "m=", 56, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8

"k=", 2, "m=", 92, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8

"k=", 3, "m=", 74, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8

"k=", 3, "m=", 92, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8





ap> From seqfan-owner at ext.jussieu.fr  Tue Apr 22 18:04:19 2008
ap> Date: Tue, 22 Apr 2008 12:03:47 -0400
ap> From: "Alexander Povolotsky" <apovolot at gmail.com>
ap> To: "Maximilian Hasler" <maximilian.hasler at gmail.com>, franktaw at netscape.net,
ap>         zakseidov at yahoo.com
ap> Subject: Re: %C A136296 "Special augmented numbers" from Zak Seidov
ap> Cc: SeqFan <seqfan at ext.jussieu.fr>
ap> ...
ap> What about relation between A136296 and A116436 ?
ap>
ap> 
ap> A116436  Numbers n which when sandwiched between two 1's give a multiple of n.
ap> 1, 11, 13, 77, 91, 137, 9091, 909091, 5882353, 10989011, 12987013,
ap> 52631579, 76923077, 90909091, 4347826087, 9090909091, 13698630137,
ap> 909090909091, 3448275862069, 10989010989011, 12987012987013,
ap> 76923076923077 (list; graph; listen)
ap> OFFSET

This is simple, even I can answer that. We may put it into a formula to let
it look more interesting than it is:

%F A136296 A116436 INTERSECT A000040 .

--
Richard Mathar