Carl Størmer and his Numbers
Matthew Kroesche, Lance L. Littlejohn, Graeme Reinhart
TL;DR
The paper precisely characterizes when a given $x_0$ can serve as the Størmer number for some prime $p \equiv 1 \pmod{4}$ by linking this to the prime-factorization of $x_0^2+1$: there exists such a prime iff the largest prime factor $p_m$ of $x_0^2+1$ satisfies $2x_0+1 \le p_m$, in which case $p=p_m$ and $S(p_m)=x_0$. It proves the injectivity of the Størmer function and discusses the implied infinitude of Størmer numbers, along with density conjectures (conjectured density $\ln 2$) via heuristic arguments. The work also connects Størmer numbers to Gregory numbers and arctan identities, showing how non-Størmer indices decompose into linear combinations of those at Størmer indices, with concrete examples that lead to famous $\pi$-related formulas and identities through Gaussian-integer factorization. A biographical sketch of Carl Størmer and the historical context of these ideas rounds out the study, highlighting the cross-disciplinary influence between number theory and the approximation of $\pi$ via Gregory/Madhava-style series.
Abstract
In many proofs of Fermat's Two Squares Theorem, the smallest least residue solution $x_0$ of the quadratic congruence $x^2 \equiv -1 \bmod p$ plays an essential role; here $p$ is prime and $p \equiv 1 \bmod 4$. Such an $x_0$ is called a Størmer number, named after the Norwegian mathematician and astronomer Carl Størmer (1874-1957). In this paper, we establish necessary and sufficient conditions for $x_0 \in \mathbb{N}$ to be a Størmer number of some prime $p \equiv 1 \bmod 4$. Størmer's main interest in his investigations of Størmer numbers stemmed from his study of identities expressing $π$ as finite linear combinations of certain values of the Gregory-MacLaurin series for $\arctan(1/x)$. Since less than 600 digits of $π$ were known by 1900, approximating $π$ was an important topic. One such identity, discovered by Størmer in 1896, was used by Yasumasa Kanada and his team in 2002 to obtain 1.24 trillion digits of $π$. We also discuss Størmer's work on connecting these numbers to Gregory numbers and approximations of $π$. \u