Designations Steinhaus - Moser

Steinhaus-Moser notation is a very large integer notation method proposed by Hugo Steinhaus and is represented using polygons.

First operations:

= n ⁿ ;
= _n - n lies in the triangle n times;
= _n - n is squared n times;

and so on.

Steinhaus himself used only three operations, the latter being designated as n in the circle:

=

.

We introduce the notation: $M(n,m,p)$ ${\ displaystyle M (n, m, p)}$ $M (n, m, p)$ - n embedded n times in a p -gon. Then you can define the rules for calculating the values of the Steinghaus - Moser polygons:

$M(n,1,3)=n^{n}$ ${\ displaystyle M (n, 1,3) = n ^ {n}}$ $M (n, 1,3) = n ^ {n}$ ,
$M(n,1,p+1)=M(n,n,p)$ ${\ displaystyle M (n, 1, p + 1) = M (n, n, p)}$ $M (n, 1, p + 1) = M (n, n, p)$ ,
$M(n,m+1,p)=M(M(n,1,p),m,p)$ ${\ displaystyle M (n, m + 1, p) = M (M (n, 1, p), m, p)}$ $M (n, m + 1, p) = M (M (n, 1, p), m, p)$ .

Respectively,

= $M(n,1,3)$ ${\ displaystyle M (n, 1,3)}$ ${\ displaystyle M (n, 1,3)}$ ;
= $M(n,1,4)$ ${\ displaystyle M (n, 1,4)}$ ${\ displaystyle M (n, 1,4)}$ ;
= $M(n,1,5)$ ${\ displaystyle M (n, 1,5)}$ ${\ displaystyle M (n, 1,5)}$

Special Values

Some numbers have special names:

mega - 2 in a circle: ② (last 14 digits: ... 93539660742656) or $M(2,1,5)$ ${\ displaystyle M (2,1,5)}$

{\begin{aligned}M(2,1,5)&=M(2,2,4)=M(M(2,1,4),1,4)=M(M(2,2,3),M(2,2,3),3)=\\&=M(M(M(2,1,3),1,3),M(M(2,1,3),1,3),3)=M(M(2^{2},1,3),M(2^{2},1,3),3)=\\&=M(4^{4},4^{4},3)=M(256,256,3)=M(256,256,3)\approx (256\uparrow )^{256}257\end{aligned}}

{\ displaystyle {\ begin {aligned} M (2,1,5) & = M (2,2,4) = M (M (2,1,4), 1,4) = M (M (2, 2,3), M (2,2,3), 3) = \\ & = M (M (M (2,1,3), 1,3), M (M (2,1,3), 1,3), 3) = M (M (2 ^ {2}, 1,3), M (2 ^ {2}, 1,3), 3) = \\ & = M (4 ^ {4} , 4 ^ {4}, 3) = M (256,256,3) = M (256,256,3) \ approx (256 \ uparrow) ^ {256} 257 \ end {aligned}}}

megiston - 10 in a circle: ⑩ or $M(10,1,5)=M(10,10,4)$ ${\ displaystyle M (10,1,5) = M (10,10,4)}$
Moser number - 2 in a megagon (polygon with mega sides), i.e. $M(2,1,M(2,1,5))=M(2,1,M(256,256,3))$ ${\ displaystyle M (2,1, M (2,1,5)) = M (2,1, M (256,256,3))}$ .

Comparing with the function that determines the Graham number , we can see that mega and megiston are less than g ₁ (the so-called "grahal", Grahal), and the brain is located between g ₁ and g ₂ .

Links

Weisstein, Eric W. Steinhaus-Moser's Notation on Wolfram MathWorld .
| The last 14 digits of the Mega number

Designations Steinhaus - Moser

Special Values

See also

Links

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