Conference rusure::math

>o	Can a number ever have TWO or more ways to get to it from 3,
>	just using factorial and integer square root ?
    
    3 = <ffssfss>

o	Can a number ever have TWO or more ways to get to it from 3,
	just using factorial and integer square root ?


This one's easy: since you have a representation for 7 and others for 49,
50 ... in the list, just add an 's' to the end of the 49th, 50th, ...) to
get a second, third... for 7. 

Lynn Yarbrough 

    ... you have a solution for 40. If you apply fsssss to it you will get
    31. Similarly if you ever find a solution to 42 (:-) you can apply
    fsssss to that and get 39 which is another gap in your table.
    
    I'm quite willing to believe that any number can be thus represented. I
    wouldn't be surprised if it was also true if you made the restriction
    that you had to do all the f's before the s's.
    
    Proof? That's another matter.
    
    dick

Several weeks ago, someone in rec.puzzles asked about generating ALL positive
integers, starting with 3, and successively taking factorial or integer square
root.

I wrote a program to try playing.  Each line in the following list tells how
to get to the given integer, starting from 3.  Each "s" means take
integer square root.  Each "f" means take factorial and then take
enough square roots to bring the result down below 100000.

For example, the line

	5 = <ffss>

means we started from 3, took the factorial, then the factorial again, then
square root twice, and got to 5.

As another example, the line

	4 = <ffssfsfsfss>

means start from 3, take factorial giving 6, take factorial giving 720,
square root twice giving 5, factorial giving 120, square root giving 10,
factorial giving 362880, square root giving 146544, factorial giving etc.
etc.

Here's the list.

See end of list for some questions:

 1 = <s>
 2 = <fs>
 3 = <>
 4 = <ffssfsfsfsfssfssfsfss>
 5 = <ffss>
 6 = <f>
 7 = <ffssfsfsfsfss>
 8 = <ffssfsfsfsfssfss>
 9 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffss>
10 = <ffssfs>
11 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffs>
12 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfs>
13 = <ffssfsfsfsfssfssfsfssffsffsffsfsfs>
14 = <ffssfsfsfsfssfssfs>
15 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffs>
16 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfs>
17 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffs>
18 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsfffsffffffsffsfsffffs>
19 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsffs>
20 = <ffssfsfsfsfssfssfsfssffsffs>
21 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsffsfs>
22 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfs>
23 = <ffssfsfsfsfssfssfsfs>
24 = <ffssfsfsfsfssfssfsfssf>
25 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffs>
26 = <ffs>
27 = <ffssfsfsfsfssfssfsfssffsffsffs>
28 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsfffsffffffsffsfsffffsfffffs>
29 = <ffsffs>
30 = <ffssfsfsfsfssfssfsfssffs>
31 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsfffs>
32 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfs>
33 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsfffsffffsfffffffs>
34 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfs>
35 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsfffsffffffsffsfffsffsfs>
36 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsfffffffs>
37 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfs>
38 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfs>
39 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffs>
40 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsfffsffffffsffsfffsffsffs>
41 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfsfffsffffffsffsfffsffs>
42 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsffs>
43 = <ffssfsfs>
44 = <ffssfsfsfs>
45 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsffffffffffffs>
46 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfs>
47 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfs>
48 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsffsffsfsffssfffffsffsfsfsffsfsfsfsffsfsfffffffffffs>
49 = <ffssfsfsfsfssfssfsfssffsffsffsfsfsfsfsfsffsfsfsfffsfsfffffffffffs>
50 = <ffssfsfsfsfs>


Questions:

o	Someone said Knuth says ALL integers can be reached with just
	factorial and integer square root, starting from 3.  Is this
	conjecture, or has it been proven ?

o	Can a number ever have TWO or more ways to get to it from 3,
	just using factorial and integer square root ?

/Eric

    
    WHOOPS!  There was a bug in my bignum factorial routine.  .4 shows
    the corrected .0.
    
    There are no more gaps. from 1-50 in the list.  Thanks to participant
    who pointed out how 31 could be produced from 40 by just doing fsss...
    
    That made me suspicious of my program, and I found a bug.
    
    /Eric

    :-)