Bowers' Arrays II

Number Theory Level 4

Evaluate log ⁡ 3 n \log _{ 3 }{ n } where n n is the value of the Bowers' array { 3 , 4 , 2 } \{ 3,4,2\} .

Hint: Remember that an array of the form { 3 , b } \{ 3,b\} = 3 b 3^b .

Note: Bowers' Arrays are a fairly obscure notation. An explanation of them can be found here.


The answer is 7.6255975E+12.

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1 solution

Ivan Koswara
Jun 28, 2014

{ 3 , 4 , 2 } = { 3 , { 3 , 3 , 2 } , 1 } \{3,4,2\} = \{3,\{3,3,2\},1\} by Rule 5

{ 3 , { 3 , 3 , 2 } , 1 } = { 3 , { 3 , 3 , 2 } } \{3,\{3,3,2\},1\} = \{3,\{3,3,2\}\} by Rule 2

{ 3 , { 3 , 3 , 2 } } = 3 { 3 , 3 , 2 } = n \{3,\{3,3,2\}\} = 3^{\{3,3,2\}} = n by Rule 1

Thus log ⁡ 3 n = log ⁡ 3 3 { 3 , 3 , 2 } = { 3 , 3 , 2 } = 3 27 \log_3 n = \log_3 3^{\{3,3,2\}} = \{3,3,2\} = \boxed{3^{27}} by Problem 1 .

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