Carl St{\o}rmer and his Numbers

TL;DR

This paper characterizes when the smallest solution to a quadratic congruence related to Fermat's Two Squares Theorem is a Størmer number, connecting number theory with pi approximations and historical identities.

Contribution

It establishes necessary and sufficient conditions for a natural number to be a Størmer number of some prime congruent to 1 mod 4.

Findings

Derived conditions for Størmer numbers related to primes p ≡ 1 mod 4.

Connected Størmer numbers to identities approximating π.

Discussed historical and computational significance of Størmer's work.

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 $\pi$.