Knuth Arrow Theorem

The Knuth Arrow Theorem is a theorem proven by Sbiis Saibian. It states that for any integers a≥2, b≥1, c≥1, x≥2, using up-arrow notation, (a↑xb)↑xc < a↑x(b+c). 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