Third Loneliest Number

Algebra Level 3

x x x 16 = 16 , x x 16 = ? \LARGE x^{x^{x^{16}}} = 16 \qquad, \qquad x^{x^{16}} = \ ?

Solve for real x x .


The answer is 16.

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Garrett Clarke
Jul 28, 2015

Please consider the following infinite power tower:

y = x x x x … y=x^{x^{x^{x^{\dots}}}}

x y = x x x x … = y x^y=x^{x^{x^{x^{\dots}}}}=y

x x y = x x x x … = y x^{x^y}=x^{x^{x^{x^{\dots}}}}=y

x x x y = x x x x … = y x^{x^{x^y}}=x^{x^{x^{x^{\dots}}}}=y

Let y = 16 y=16 and we have the identity x x x 16 = x x 16 = 16 x^{x^{x^{16}}}=x^{x^{16}}=\boxed{16}

15 Helpful 0 Interesting 0 Brilliant 0 Confused
A Sad Yam
Dec 31, 2017

Since x^16 = 16 = what is given in the problem because of chain cancellation we have a chain which is also cancelled, but in a different factor. In power towers, the number of x's never matter because of cancellation therefore the answer is still 16.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

No. If I replace all the numbers 16 by another number (say 12), then the answer would not be 12.

Pi Han Goh - 3 years, 5 months ago

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x^x^x^x^...^12 = 12 -> x^12 = 12 so x = 12^(1/12) therefore in any case of x^(any n)^12 =a, a = 12.

A Sad Yam - 3 years, 5 months ago

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