Pi Formulas: some smooth stones on the beach of rough numbers

Abstract

This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set $S_k$ of rough numbers, where $k$ is a prime number. Given $S_k$ for any prime $k$, the set $S_{k^{\prime}}$, where $k^{\prime}$ is the smallest prime greater than $k$, can be constructed easily. From this it follows that Pi Formulas occur in disjoint families. In any family, there is a first member, a series of least prime number $k_{min}$, which must be summed from first principles. Then from the series of some $k > k_{min}$ already summed, the series for $k^{\prime}$ can be summed by a simple algebraic procedure. A good number of Pi Formulas, belonging to a variety of families and giving results believed to be new, are presented here.