This paper introduces a new class of "unbounded" spigot algorithms for calculating the decimal digits of $pi$, improving upon the classic Rabinowitz–Wagon method. While previous spigot algorithms required users to commit to a specific number of digits in advance and faced potential errors due to carry-over effects from truncated series, this proposed approach eliminates those limitations by allowing for infinite digit generation given sufficient memory. Although not intended to compete with high-performance state-of-the-art arithmetic-geometric mean algorithms, the author’s method offers a mathematically robust, simple, and incrementally efficient way to produce digits one by one without prior commitment or risk of truncation errors.
