mathematics

Hᵤ(2, 2) = 4 for all u>0

A bit of recreational mathematics. I’d be (pleasantly) shocked if this is novel.

2+2 = 2×2 = 2² = 4 🤔

Hᵤ is the set of hyperoperations, e.g.
H₁(a, b) = a + b
H₂(a, b) = a × b

Hu(a, b) = Hu-1(a, Hu(a, b-1)) when u≥3 & b≠0
∴ Hu(2, 2) = Hu-1(2, Hu(2, 1)) when u≥3

also Hu(2, 1) = Hu-1(2, Hu(2, 0)) when u≥3

also Hu(2, 0) = 1 when u≥3

∴ Hu(2, 1) = Hu-1(2, 1) when u≥3
∴ H3(2, 1) = H2(2, 1) = 2
∴ H3(2, 2) = H2(2, 2) ⇒ 2² = 2 × 2 = 4

also Hu(2, 1) = 2 when u≥3
∴ Hu(2, 2) = Hu-1(2, 2) when u≥3
∴ ∀ u ≥ 3 Hu(2, 2) = 4

as H₁(2, 2) = 4 and H₂(2, 2) = 4, ∀ u ≥ 1 Hu(2, 2) = 4

i.e., for any hyperoperation ∘ other than the zeroth (successor) operator, 2 ∘ 2 = 4

Standard

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