[seqfan] Re: "What primes P have P^P containing the string 'P' as substring'?"

[Charles Greathouse]

p^p has about p log_b p base-b digits, while p has about log_b p
base-b digits.  The chance that a given substring of length log_b p
would have an arbitrary value is b^-log_b p, so the chance that such a
substring would not have that value is
1 - b^-log_b p
and the chance that none of the ~ p log_b p length-log_b p substrings
of p^p would have that value is
(1 - b^-log_b p)^(p log_b p)
which is
(1 - 1/p)^(p log_b p) = ((1 - 1/p)^p)^(log_b p)
which is asymptotically
(1/e)^(log_b p) = p^(-1/ln b)

For base 10, that is something like x^0.5657 not of that form below x.

I count 440 of that form and 229 not of that form below 5000.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Apr 26, 2010 at 4:07 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> JM Bergot asked me in an email: "What primes P have P^P containing the
> string 'P' as substring'?"
>
> 5^5 = 3125 has "5" as substring.
>
> 11^11 = 285311670611 has two substrings of "11"
>
> 17^17 = 827240261886336764177  has "17" as 3rd and 2nd digit from right-hand.
>
> 19^19 = 1978419655660313589123979 has a "19" as its left end, and another later.
>
> 31^31 = 17069174130723235958610643029059314756044734431
>
> has a "31" at right-hand end, and a second to the left a ways.
>
> A051674 (n-th prime)^(n-th prime).
>
> Is there a sequence in there, base and prime related, struggling to be
> interesting to anyone on seqfans?
>
> Eventually, all P^P are pandigital.  But not "normal." Right?
>
> Best,
>
> Jonathan Vos Post
>
>
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