# Self-powers numbers (SPN)

Graeme McRae g_m at mcraefamily.com
Tue Jul 5 18:32:25 CEST 2005

```OK, then, since you've proved uninteresting the task of finding all such
SPN, what about finding (and proving) the smallest PSPN
(pan-self-power-number)?  2799363256438443359375 is such a number, of
course, consisting of three parts.

279936 is the smallest seventh power that does not contain any zeros or
ones.
38443359375 is the smallest ninth power that does not contain any zeros or
ones.
Between them, in the middle of this string, 32564 is particularly efficient
at the task of disposing of the square, cube, 4th, 5th, 6th, and 8th powers.

The only hope of finding a smaller PSPN would lie in finding a 7th and 9th
power that share some digits with each other or with 32564.  An exhaustive
search would suffice, because 1159^7 and 242^9 both exceed
2799363256438443359375.

--Graeme

----- Original Message -----
From: "Jack Brennen" <jb at brennen.net>
To: <seqfan at ext.jussieu.fr>
Sent: Tuesday, July 05, 2005 7:37 AM
Subject: Re: Self-powers numbers (SPN)

> Eric Angelini wrote:
> > Question:
> > Can someone compute all such SPN _which
> > don't include any 0's or 1's_ ?
>
> There are an infinite number of course...
>
> For example, any sequence of digits 2-9 which contains this
> string:
>
>    2799363256438443359375
>
> Showing this is an SPN is left as an exercise for the reader.
>
>

```