EDITED - sequences related to x^x+p being prime

[cino hilliard]

Hi Dean,

Thanks for all your work.

>From: Dean Hickerson <dean at math.ucdavis.edu>
>To: njas at research.att.com
>CC: hillcino368 at hotmail.com
>Subject: EDITED - sequences related to x^x+p being prime
>Date: Tue, 29 Jul 2003 02:10:20 -0700
>
>Neil:  Please delete sequences A086654, A086655, and A086656 from Cino
>Hilliard.
>
>For reference, here are the current %S and %N lines of these:
>
>%S A086654 
>5,8,11,17,24,41,50,53,59,65,77,83,89,90,98,118,119,125,129,142,143,144,
>%N A086654 Numbers p such that x^x + p produces no primes for x = 1,2,..
>
>
>%S A086655 
>1,2,3,4,6,7,9,10,12,13,14,15,16,18,19,20,21,22,23,25,26,27,28,29,30,31,
>%N A086655 Numbers p such that x^x + p produces primes for x = 1,2,..
>
>
>%S A086656 
>1,2,4,1,3,2,1,3,7,43,1,2,4,6,32,2,1,3,7,9,39,1,5,2,4,16,42,3,23,2,4,1,
>%N A086656 Numbers x such that x^x + p is prime for p = 1,2,..
>
>
>A086654 is defined as "Numbers p such that x^x + p produces no primes for
>x = 1,2,.." and begins with 5,8,11.  But Mathematica reports that x^x+5 is
>probably prime for x=444.  I haven't found probable primes of the forms
>x^x+8 and x^x+11, but I see no reason to doubt their existence:
>Heuristically, the probability that x^x+8 is prime is about 1/(x log(x)).
>The sum of this from x=1 to n is about log(log(n)), which goes to infinity
>as n does.  So it seems likely that there are infinitely many primes of the
>given form unless there's some "obvious" reason to the contrary.  I don't
>see any such reason, so I suspect that there are infinitely many primes of
>the given form.  The same is true for the form x^x+p for any p>1, so I
>believe that sequence A086654 is empty.  (p=1 is different:  x^x+1 can only
>be prime if x=1 or x has the form 2^2^r with r>=0 and x^x+1 = 2^2^(r+2^r)+1
>is a Fermat prime.  There probably aren't any more of these besides x=1,
>2, and 4.)
>
>A086655 appears to be the complement of A086654, so if that sequence is
>empty, then this one consists of all positive integers.
>
>A086656 seems to be defined as the concatenation of the set of x for which
>x^x+1 is prime, the set of x for which x^x+2 is prime, ...; it begins with
>1,2,4, 1,3, 2, 1,3,7,43 because
>
>    x^x+1 is prime for x = 1, 2, 4;
>    x^x+2 is prime for x = 1, 3;
>    x^x+3 is prime for x = 2;
>    x^x+4 is prime for x = 1, 3, 7, 43.
>
>As I said above, it seems likely that the above list for x^x+1 is complete.
>But I think each of the other lists is infinite, so we can't concatenate
>them.  (In particular, x^x+2 is probably prime for x=737, and x^x+9 is
>probably prime for x=130 and x=140; these aren't shown in the current
>sequence.)
>
>Dean Hickerson
>dean at math.ucdavis.edu

I tried isprime(444^444+5) on Gp 2.2.6 alpha and it bombed. I cc'd you  My 
Email  report to Karim
Belabas, the Pari developer.

Too bad we will never know if 444^444+5 is _really_  a prime.

This has roused my curiousity. How probable is a probable prime in 
Mathematica,Maple,Pari? I assume
they all use the same algorithm but what is the number? 1 in 10^10, 1 in 
10^100? 1 in googolplex
that it is not really prime?  I've got it!

                    99 and 444 one thousands percent pure?

Thanks,
Cino Hilliard
"Observation lends to analysis. Analysis lends to Law."

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