Carl Størmer and his Numbers

Abstract

In many proofs of Fermat's Two Squares Theorem, the smallest least residue solution $x_0$ of the quadratic congruence $x^2\equiv -1\mathrm{mod}p$ plays an essential role; here $p$ is prime and $p\equiv 1\mathrm{mod}4$. 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 $p\equiv 1\mathrm{mod}4$. Størmer's main interest in his investigations of Størmer numbers stemmed from his study of identities expressing $\ddot{I}€$ as finite linear combinations of certain values of the Gregory-MacLaurin series for $\arctan (1/x)$. Since less than 600 digits of $\ddot{I}€$ were known by 1900, approximating $\ddot{I}€$ 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 $\ddot{I}€$. We also discuss Størmer's work on connecting these numbers to Gregory numbers and approximations of $\ddot{I}€$.