Carl St{\o}rmer and his Numbers

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 St{\o}rmer number, named after the Norwegian mathematician and astronomer Carl St{\o}rmer (1874-1957). In this paper, we establish necessary and sufficient conditions for $x_0 \in \mathbb{N}$ to be a St{\o}rmer number of some prime $p \equiv 1 \bmod 4$. St{\o}rmer's main interest in his investigations of St{\o}rmer numbers stemmed from his study of identities expressing $\pi$ as finite linear combinations of certain values of the Gregory-MacLaurin series for $\arctan(1/x)$. Since less than 600 digits of $\pi$ were known by 1900, approximating $\pi$ was an important topic. One such identity, discovered by St{\o}rmer in 1896, was used by Yasumasa Kanada and his team in 2002 to obtain 1.24 trillion digits of $\pi$. We also discuss St{\o}rmer's work on connecting these numbers to Gregory numbers and approximations of $\pi$.