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Knuth's Arrows

Tool to write with Arrowed notation of iterative exponentiation by Knuth : a mathematical notation whose purpose is to write huge integer numbers with powers.

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Knuth's Arrows -

Tag(s) : Mathematics, Notation System

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# Knuth's Arrows

## Calculation with Knuth's up-arrows notation A↑↑B

Tool to write with Arrowed notation of iterative exponentiation by Knuth : a mathematical notation whose purpose is to write huge integer numbers with powers.

### How to calculate using Knuth up-arrows notation?

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

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 afford to display without risking blocking your browser, so there's a limit of 100,000 digits.

### What does 3 Knuth up-arrows means?

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