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  • when we add the same number many times we can use multiplication for shorter: $$2+2+2+2 = 2\times4$$
  • when we multiply the same number many times we can use exponentiation for shorter: $$2\times2\times2\times2 = 2^4$$
  • is there next level of shorter form equivalent to rising to the same power over and over again? $$2^{2^{2^2}} = ?(2,4)$$ In the above example there is not much of shortening because we can mulitply powers and have $$2^8 = ?(2,4)$$ But look at this: $$1000^{1000^{1000^{1000^{1000^{1000}}}}} = 1000^{1000000000000000} = ?(1000,6)$$
  • if such a operation exists what about next levels of shortening?
Tomek Wyderka
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    Welcome to Mathematics Stack Exchange. Cf. tetration – J. W. Tanner Oct 20 '21 at 03:42
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    Remember that $a^{b^c}=a^{(b^c)}$ and not $((a^b)^c)$. So $2^{2^{2^{2}}}\neq 2^8$. – Sebastián P. Pincheira Oct 20 '21 at 03:52
  • I'd suggest starting from the WP article on tetration, which is to exponentiation as exponentiation is to multiplication. That will also lead to to Knuth up-arrow notation. – Eric Snyder Oct 20 '21 at 05:38
  • And as noted, $2^{2^{2^{2}}} \neq 2^8$, and is instead $2^16$. These functions are sometimes called "hyperoperations," represented as $H_n(a,b)$, where $a$ is the base and $b$ is the equivalent of an exponent. The $n$ subscript is the "level" of hyperoperations: e.g., $H_1(2,3)= 2 \times 3, H_4(2,2)$ is the example above, etc. – Eric Snyder Oct 20 '21 at 05:41

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I believe that you're after something like Knuth's up-arrow notation. This has been explained on this site. See How exactly does Knuth's Up-Arrow notation work?

Auslander
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