Knuth Arrow Theorem

The Knuth Arrow Theorem is a theorem proven by Blakley and Borosh in 1979.Blakley, G.R., Borosh, I.: Knuth’s iterated powers. Adv. Math. 34(2), 109–136 (1979). https://doi.org/10.1016/0001-8708(79)90052-5 Theorem 3.1 p. 120 The theorem is described as
For any integers \(x \ge 2, a \ge 2, b \ge 1, c \ge 1\), we have
\[a \uparrow ^x(b+c-1) \le (a \uparrow ^x b) \uparrow ^xc < a \uparrow ^x(b+c)\]
where equality holds on the left side if and only if \(b=1\) or \(c=1\).
Sbiis Saibian independently rediscovered the right-hand part of this theorem in 2014.Sbiis Saibian A Theorem for Knuth Arrows
The theorem is useful for comparing values in up-arrow notation. For example, it can be used to prove that giggol (10↑↑100) is smaller than tritri (3↑↑↑3 = 3↑↑7,625,597,484,987):
10↑↑100
< 27↑↑100
= (3↑↑2)↑↑100
< 3↑↑102
<< 3↑↑7,625,597,484,987 = tritri
== See also ==
* Bird's Proof
* Arrow notation
== References ==
Category:Theorems