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?