Carl Størmer and his Numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917446347128832 |
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| author | Kroesche, Matthew Littlejohn, Lance L. Reinhart, Graeme |
| author_facet | Kroesche, Matthew Littlejohn, Lance L. Reinhart, Graeme |
| contents | 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 $π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03030 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Carl Størmer and his Numbers Kroesche, Matthew Littlejohn, Lance L. Reinhart, Graeme History and Overview Number Theory 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 $π$. |
| title | Carl Størmer and his Numbers |
| topic | History and Overview Number Theory |
| url | https://arxiv.org/abs/2511.03030 |