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Quintation

Quintation is the fifth step in hyper operators. It is iterated tetration. It follows the sequence after tetration:

addition
$a + b$
multiplication
${{\ } \atop a \times b = } {{b\mbox{ copies of }a} \atop {\overbrace{a + \cdots + a}}}$
exponentiation
${{\ } \atop a^b = } {b\mbox{ copies of }a \atop {\overbrace{a \times \cdots \times a}}}$
tetration
${\ ^ba = \atop {\ }} \!\!\!\!\!\!\!{{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop {b\mbox{ copies of }a}}$
quintation
${\ ^b{\ ^ba = \atop {\ }}} \!\!\!\!\!\!\!{{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop {\ ^ba\mbox{ copies of }a}}$

Following the pattern with tetration ( $a \uparrow\uparrow b$ ) being b instances of a exponentated together, quintation ( $a \uparrow\uparrow\uparrow b$ ) is b instances of a tetrated together, or ^ba instances of a exponentiated together.

Using Knuth's up-arrow notation, you can simplify quintation like this:

$3\uparrow\uparrow\uparrow2 = 3\uparrow\uparrow3 = 3\uparrow3\uparrow3 = 3^{27};$
$2\uparrow\uparrow\uparrow3 = 2\uparrow\uparrow(2\uparrow\uparrow2) = 2\uparrow\uparrow4 = 65,536;$
$3\uparrow\uparrow\uparrow3 = 3\uparrow\uparrow(3\uparrow\uparrow3) = 3\uparrow\uparrow(3\uparrow3\uparrow3) = 3\uparrow\uparrow3^{27} = 3\uparrow3\uparrow\cdots\uparrow3 \quad (3^{27}\mbox{ copies of } 3);$
$2\uparrow\uparrow\uparrow4 = 2\uparrow\uparrow(2\uparrow\uparrow(2\uparrow\uparrow2)) = 2\uparrow\uparrow(2\uparrow\uparrow4) = 2\uparrow\uparrow65536.$

Even if you quintate to power 3, the numbers grow very fast.

Also, if you come across a quintation in form $a \uparrow\uparrow\uparrow2$ , you can change it to $a \uparrow\uparrow a$ as seen in the $3\uparrow\uparrow\uparrow2$ simplification above.

Values

This operation has only a few non-trivial cases, such as $3\uparrow\uparrow\uparrow2$ = 7,625,597,484,987, that produce numbers that can be written in conventional notation. There are others:

$1\uparrow\uparrow\uparrow1$ = 1
$1\uparrow\uparrow\uparrow2$ = 1
$1\uparrow\uparrow\uparrow3$ = 1 etc.
$2\uparrow\uparrow\uparrow2$ = 4
$3\uparrow\uparrow\uparrow2$ = 7,625,597,484,987

Science Fair Project Encyclopedia Contents Page

11-30-2008 18:11:33
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