--- In

primenumbers@yahoogroups.com, "David Broadhurst" <d.broadhurst@...> wrote:

> Like Richard, I find this result to be "strange indeed".

>

> Can anyone tell Richard and me why we should not be surprised?

I'm still working on the "why". I suspect it has something to do with how quickly iterates of eulerphi for a prime hit a power of 2. The phenomenon doesn't seem to be rare though.

For David's example the numbers were 137 and 73.

I use 139 and 15 selected by trying a few numbers until lots of small factors showed up at higher iterations.

139+15 has factors 2, 7, 11

139^139+15 has factors 2, 7, 19, 8689, 60293

139^139^139+15 has factors 2, 7, 19, 67, 983, 1723, 66841

139^139^139^139+15 has factors 2, 7, 19, 67, 1723, 66841

139^139^139^139^139+15 has factors 2, 7, 19, 67, 1723, 66841

139^^6+15 has factors 2, 7, 19, 67, 1723, 66841

139^^7+15 has factors 2, 7, 19, 67, 1723, 66841

139^^8+15 has factors 2, 7, 19, 67, 1723, 66841

139^^9+15 has factors 2, 7, 19, 67, 1723, 66841

etc

So some primes pop in for an interation and go again and some come in and stick around.

7 keeps showing up as 139 % 7 = -1 and 15 % 7 = 1

19 keeps showing up as eulerphi(eulerphi(19))=6 and 139 % 6 = 1

67 keeps showing up as eulerphi(eulerphi(67))=20 and 139 % 20 =-1

I can't quite make sense of 1723 and I've run out of time.

Richard Heylen