Tower of Powers

Consider an infinite “tower of powers” of x, defined by

x^x^x^… = x^(x^(x^…))

Can we find a value of x so that this tower is equal to 2? Let’s write A for the value of the tower; then also A = xA from the definition, so solving for x we get x = A1/A. If A = 2, then x = Sqrt[2].

How about A = 4? For this one we get

x = 41/4 = sqrt(2),

the same as for A = 2. In words, we conclude that the tower of powers

Sqrt(2)^Sqrt(2)^Sqrt(2)… = both 2 and 4.

Which is correct?

Presentation Suggestions:
Present as above and ask the students as homework to think of explanations for this strange behavior. At some point they will discover (or need to be shown) that the tower can be defined as the limit of a sequence: x, x^x, x^x^x, …; they should then try calculating numerically the limit for x = Sqrt(2) to see if they can decide which answer, 2 or 4, is correct.

The Math Behind the Fact:
Students need some background in limits to understand this, since the critical point is that the tower has to be defined somehow; we can’t just write down an of symbols and assume that it has a meaning. (Euler could do this, but we can’t.) It turns out that the equation has no solutions for A > e, which has a simple graphical explanation once the is understood. This example also illustrates the hazards of calculating a “solution” to an equation without knowing that a solution exists. The article by Mitchelmore has good diagrams and a careful consideration of all the cases.

How to Cite this Page: 
Su, Francis E., et al. “Tower of Powers.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
“A Matter of Definition” by M. C. Mitchelmore.
in A Century of , Part II edited by Tom Apostol et al., pp. 84-88. Reprinted from the Amer. Math. Monthly, 81(1974), 643-647.

Fun Fact suggested by:
Allen Stenger

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