Knuth's Arrows

The Knuth arrows are a repeated exponentiation representation (Knuth up arrows). As multiplication is the repetition of additions ($ 2 \times 3 = 2+2+2 $), as exponentiation is the repetition of multiplications ($ 2^3 = 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 (a single arrow represents an exponentiation)

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

Knuth's notation with 2 arrows is an iterated power

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

Example: $$ 3 \uparrow\uparrow 2 = 3^3 = 27 \\ 3 \uparrow\uparrow 3 = 3^{3^3} = 3^{27} = 7625597484987 \\ 3 \uparrow\uparrow 4 = 3^{3^{3^3}} = 3^{3^{27}} = 3^{7625597484987} $$

It may be noted that,

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

Example: $$ 3 \uparrow\uparrow 2 = 3 \uparrow 3 = 3^3 \\ 3 \uparrow\uparrow 3 = 3 \uparrow 3 \uparrow 3 = 3^{3^3} $$

Knuth's arrows produce immensely large numbers (big integers) that dCode can not display without risking blocking your browser, so there's an automatic limit above several thousands of digits.

The 3 arrows (triple arrow) notation is the continuity of the 2 arrows notation (double arrow)

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

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

Knuyth's arrows make it possible to represent numbers so large that the usual notations do not allow them to be described simply or precisely.

The dCode calculator is therefore limited, because the numbers of iterations quickly exceed the capacities of the computers...