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Goodstein Sequence

Given a Hereditary Representation of a number $n$ in Base, let $B[b](n)$ be the Nonnegative Integer which results if we syntactically replace each $b$ by $b+1$ (i.e., $B[b]$ is a base change operator that `bumps the base' from $b$ up to $b+1$). The Hereditary Representation of 266 in base 2 is

$\displaystyle 266$ $\textstyle =$ $\displaystyle 2^8+2^3+2$  
  $\textstyle =$ $\displaystyle 2^{2^{2+1}}+2^{2+1}+2,$  

so bumping the base from 2 to 3 yields


Now repeatedly bump the base and subtract 1,
$\displaystyle G_0(266)$ $\textstyle =$ $\displaystyle 266=2^{2^{2+1}}+2^{2+1}+2$  
$\displaystyle G_1(266)$ $\textstyle =$ $\displaystyle B[2](266)-1=3^{3^{3+1}}+3^{3+1}+2$  
$\displaystyle G_2(266)$ $\textstyle =$ $\displaystyle B[3](G_1)-1=4^{4^{4+1}}+4^{4+1}+1$  
$\displaystyle G_3(266)$ $\textstyle =$ $\displaystyle B[4](G_2)-1=5^{5^{5+1}}+5^{5+1}$  
$\displaystyle G_4(266)$ $\textstyle =$ $\displaystyle B[5](G_3)-1=6^{6^{6+1}}+6^{6+1}-1$  
  $\textstyle =$ $\displaystyle 6^{6^{6+1}}+5\cdot 6^6+5\cdot 6^5+\ldots+5\cdot 6+5$  
$\displaystyle G_5(266)$ $\textstyle =$ $\displaystyle B[6](G_4)-1$  
  $\textstyle =$ $\displaystyle 7^{7^{7+1}}+5\cdot 7^7+5\cdot 7^5+\ldots+5\cdot 7+4,$  

etc. Starting this procedure at an Integer $n$ gives the Goodstein sequence $\{G_k(n)\}$. Amazingly, despite the apparent rapid increase in the terms of the sequence, Goodstein's Theorem states that $G_k(n)$ is 0 for any $n$ and any sufficiently large $k$.

See also Goodstein's Theorem, Hereditary Representation

© 1996-9 Eric W. Weisstein