Knuth's Arrows

The Knuth arrows are a repeated exponentiation representation. As multiplication is the repetition of additions ( \( 2 \times 3 = 2+2+2 \) ), as exponentiation is the repetition of multiplications (\ = 2 \times 2 \times 2 \)), the knuth arrows is the repetition of exponentiations (also called iterated exponentiation).

Knuth's notation with a single arrow represents a simple power operation

Example: $$ 3 \uparrow 3 = 3^3 = 27 $$

Knuth's notation with 2 arrows is an iterated power

$$ a \uparrow \uparrow b = \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}}_{ b\mbox{ times}} $$

Example: $$ 3 \uparrow\uparrow 2 = 3^3 = 27 $$

Example: $$ 3 \uparrow\uparrow 3 = 3^{3^3} = 3^{27} = 7625597484987 $$

Example: $$ 3 \uparrow\uparrow 4 = 3^{3^{3^3}} = 3^{3^{27}} = 3^{7625597484987} $$

It may be noted that

$$ a \uparrow\uparrow b = \underbrace{a_{}\uparrow a\uparrow\dots\uparrow a}_{ b\mbox{ times}} $$

Example: $$ 3 \uparrow\uparrow 2 = 3 \uparrow 3 $$

Example: $$ 3 \uparrow\uparrow 3 = 3 \uparrow 3 \uparrow 3 $$

Knuth's arrows produce immensely large numbers that dCode can not display without risking blocking your browser, so there's a limit of 100,000 digits.

the 3 arrows notation is the continuity of the 2 arrows notation

$$ a \uparrow\uparrow\uparrow b = \underbrace{a_{}\uparrow\uparrow a\uparrow\uparrow\dots\uparrow\uparrow a}_{ b\mbox{ times}} $$

Example: $$ 3 \uparrow\uparrow\uparrow 3 = 3 \uparrow\uparrow(3 \uparrow\uparrow 3) = 3 \uparrow\uparrow( 3 \uparrow 3 \uparrow 3) $$