Carrying on in the spirit of '86 at most' here is a pair of 'statistical' conjectures about 3^n: (see A060956 in Sloane, particularly the table, for the values up to 3^1000)
To start with a definition; if the first digit of 3^n is a 9 ( e.g. 3^23 = 94,143,178,827) then we say that 3^n is 'good'.
Conjecture 1: If 3^n is good, then either 3^(n+21), or 3^(n+23) is also good.
Conjecture 2: If 3^n is good, then 3^(m+n) is also good, for some constant, m, and n greater than 2.
True or false?
(In reply to
solution by Charlie)
In part 2, my analysis was a bit muddled.
The approximations should have shown closer and closer to 9, on the high side (which it did) and closer and closer to 10 but on the low side, such as 9.9999.... (some finite but lowering amount each time), which it did not.
Then the last paragraph should have said:
Any given single value for m will have its power of 3 have either a significand some finite amount above 1 or below 1 (or, rather, conventionally, below 10). Either way, it will destroy the goodness of some number that's sufficiently close to but below 1 or close to and above 9, respectively, and this sufficient degree of closeness is always achievable.
Edited on August 24, 2012, 10:07 pm

Posted by Charlie
on 20120824 22:05:37 