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Where 2^n leads, 3^n cannot be far behind... (Posted on 2012-08-24) Difficulty: 3 of 5

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?

See The Solution Submitted by broll    
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  Subject Author Date
re: solution -- correctionCharlie2012-08-24 22:05:37
SolutionsolutionCharlie2012-08-24 15:33:51
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