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

$$266 = 2^8+2^3+2$$

$$= 2^{2^{2+1}}+2^{2+1}+2,$$

so bumping the base from 2 to 3 yields

$$B[2](266)=3^{3^{3+1}}+3^{3+1}+3.$$

Now repeatedly bump the base and subtract 1,

$$G_0(266) = 266=2^{2^{2+1}}+2^{2+1}+2$$

$$G_1(266) = B[2](266)-1=3^{3^{3+1}}+3^{3+1}+2$$

$$G_2(266) = B[3](G_1)-1=4^{4^{4+1}}+4^{4+1}+1$$

$$G_3(266) = B[4](G_2)-1=5^{5^{5+1}}+5^{5+1}$$

$$G_4(266) = B[5](G_3)-1=6^{6^{6+1}}+6^{6+1}-1$$

$$= 6^{6^{6+1}}+5\cdot 6^6+5\cdot 6^5+\ldots+5\cdot 6+5$$

$$G_5(266) = B[6](G_4)-1$$

$$= 7^{7^{7+1}}+5\cdot 7^7+5\cdot 7^5+\ldots+5\cdot 7+4,$$

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