[seqfan] Re: 13532385396179 precursors

[Hans Havermann]

> Think about the number 13^532385396179. Take some of its initial decimal digits that make those digits prime and raise this to the power of the rest of the digits. Now you have a new (larger) number. Repeat.

There should be little doubt of the infinity of 13532385396179 precursors, even if the above argument is not 100% tight. The 13^532385396179 is just one of 57 primary precursors (the number 13532385396179 is excluded here from the 58 ways in which one may validly insert multiplications and powers into itself) so there are many other fall-back numbers that can extend the precursors-of-precursors situation. I've spent some time computing the the actual numbers of secondary precursors (precursors of our 57 primary precursors) and have just finished a two-week run on one of them (the 5938-digit 13^5323*853*96179, #15 in our list of 57), using a python program kindly supplied by Robert Gerbicz.

The program suggests that there are 712489702263479556568873269786312097959069025706353865676039092080541767 precursors of this number! One is not to assume that just because our primary precursor has a large number of digits it must necessarily have a huge number of secondary precursors. For example, the 59311-digit 13^53238*5396179, #26 in our list) has only 2 precursors, but this is one of only a small number of exceptions. To visualize the growth of these counts, I've put a table here:

http://chesswanks.com/num/SecondaryPrecursorCounts.txt

I don't know how much further I will get. The 411334-digit #41 has a chance of having only a small count (which I'm actively exploring). Finally, one of the secondary precursors is small enough for me to have charted its 34818 tertiary precursors:

http://chesswanks.com/num/precursors3.txt

The <number> column shows the number of decimal digits in the expressions' decimal expansions.