Gödel beta-function sequence coding
= Gödel beta-function sequence coding
{c}
The Gödel beta function codes finite sequences by remainders:
$$
\beta(b,c,i)=r
\quad\Longleftrightarrow\quad
r<1+(i+1)c\ \land\ \exists q\leq b\,[b=q(1+(i+1)c)+r].
$$
Choosing $c$ sufficiently large and divisible by the relevant small integers makes the moduli pairwise coprime, so the <Chinese remainder theorem> codes any prescribed finite sequence.