Carl St{\o}rmer and his Numbers

Graeme Reinhart; Lance L. Littlejohn; Matthew Kroesche

arxiv: 2511.03030 · v3 · submitted 2025-11-04 · 🧮 math.HO · math.NT

Carl St{o}rmer and his Numbers

Matthew Kroesche , Lance L. Littlejohn , Graeme Reinhart This is my paper

Pith reviewed 2026-05-18 00:56 UTC · model grok-4.3

classification 🧮 math.HO math.NT

keywords Størmer numbersFermat two squares theoremquadratic congruencepi approximationsGregory numbersCarl Størmerarctan series

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The pith

Natural numbers qualify as Størmer numbers exactly when they satisfy necessary and sufficient conditions tied to primes congruent to 1 mod 4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to characterize precisely which natural numbers can appear as the smallest positive solution x0 to the quadratic congruence x squared congruent to negative one modulo a prime p that is congruent to one modulo four. These solutions, called Størmer numbers, enter many proofs of Fermat's theorem on sums of two squares and figured in Størmer's own work on finite arctangent combinations that approximate pi. Establishing the conditions lets one recognize or generate such numbers directly from arithmetic properties rather than by testing primes one by one. A reader interested in classical number theory would see this as a bridge between the existence proof for the congruence and the practical identities Størmer used to push the known digits of pi far beyond what was possible in 1900.

Core claim

We establish necessary and sufficient conditions for a natural number x0 to be a Størmer number of some prime p congruent to 1 modulo 4, where x0 denotes the smallest positive integer satisfying the quadratic congruence x squared congruent to negative one modulo p.

What carries the argument

The smallest positive least-residue solution x0 to the congruence x squared congruent to negative one modulo p for primes p congruent to 1 mod 4.

If this is right

The conditions allow direct verification that a given natural number arises as a Størmer number without searching over candidate primes.
The characterization clarifies how Størmer numbers connect to Gregory numbers and to finite linear combinations of Gregory-MacLaurin arctangent series used for pi approximations.
It accounts for the numerical utility of one such identity in obtaining more than a trillion digits of pi in 2002.
Størmer's historical investigations of these numbers receive a modern arithmetic framing that links them to Fermat's two-squares theorem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The conditions may streamline the search for new arctangent identities capable of even higher-precision pi calculations.
Similar modular characterizations could apply to other quadratic residues that appear in classical proofs of prime factorization results.
Checking the conditions against small natural numbers offers a quick way to produce explicit lists of Størmer numbers for further study.

Load-bearing premise

The standard existence of a unique smallest positive solution x0 to the quadratic congruence for every prime p congruent to 1 mod 4 is taken as given.

What would settle it

A natural number that satisfies the paper's stated conditions yet fails to be the smallest solution for any prime p congruent to 1 mod 4, or a known Størmer number that violates those conditions.

Figures

Figures reproduced from arXiv: 2511.03030 by Graeme Reinhart, Lance L. Littlejohn, Matthew Kroesche.

**Figure 1.** Figure 1: Carl Størmer 4. A Biographical Sketch of Carl Størmer Carl Størmer (1874-1957) was a preeminent Norwegian mathematician, botanist, and astrophysicist. For his lifetime contributions to mathematics and astronomy, particularly his study of the aurora borealis and the motion of charged particles in magnetic fields, he was bestowed several honors. A crater on the far side of the Moon is named after him. He wa… view at source ↗

read the original 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 $\pi$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives necessary and sufficient conditions for Stormer numbers and adds historical context on pi work, but the sufficiency claim may miss the required size bound p > 2x0.

read the letter

The core contribution is a set of necessary and sufficient conditions on a natural number x0 so that it serves as the smallest positive solution to x^2 ≡ -1 mod p for some prime p ≡ 1 mod 4. The authors also sketch Stormer's original interest in these numbers through his Gregory-MacLaurin series identities for pi and note the later use by Kanada's team. That historical thread is the part that lands cleanly: it shows why anyone cared about the smallest residue in the first place and connects the definition to concrete computation history without overclaiming impact. The conditions themselves appear to rest on standard facts about quadratic residues and the factorization of x0^2 + 1, which is reasonable ground to stand on. Credit for making the characterization explicit rather than leaving it implicit in older proofs of Fermat's two-squares theorem. The potential soft spot is exactly the one flagged in the stress test. Sufficiency needs an explicit check that any prime factor p ≡ 1 mod 4 of x0^2 + 1 satisfies p > 2x0; otherwise the residue is not minimal. The counter-example with x0 = 3 and p = 5 works because 5 divides 10 yet the true least solution mod 5 is 2. If the paper's stated conditions omit or fail to enforce that inequality in the sufficiency direction, the characterization does not fully hold. Necessity looks less exposed. The paper is aimed at readers who already follow the history of pi approximations or the role of quadratic residues in classical number theory. It is narrow but self-contained, so a serious referee could check the conditions against the definition in a few pages and either confirm or tighten the size constraint. I would send it to review rather than desk reject; the claim is concrete enough to be worth verifying.

Referee Report

1 major / 2 minor

Summary. The paper defines Størmer numbers as the smallest positive integer solutions x0 to x² ≡ -1 (mod p) for primes p ≡ 1 (mod 4). It claims to establish necessary and sufficient conditions on x0 ∈ ℕ for it to be a Størmer number associated to some such prime. The manuscript also surveys Carl Størmer's historical investigations of these numbers in connection with Gregory-Maclaurin series for arctan(1/x) and their role in early 20th-century and later high-precision computations of π.

Significance. A correct characterization of Størmer numbers would clarify the arithmetic conditions underlying a standard ingredient in many proofs of Fermat's two-squares theorem and would document an under-studied thread in the history of π approximation. The historical sections supply useful context on Størmer's 1896 identity and its later computational use.

major comments (1)

[statement of necessary and sufficient conditions] The central claim (abstract and main theorem) asserts necessary and sufficient conditions for x0 to be the minimal positive solution of x² ≡ -1 (mod p). Sufficiency therefore requires that any prime p ≡ 1 (mod 4) dividing x0² + 1 satisfy p > 2x0. The manuscript must explicitly state and prove this size constraint; without it, the conditions are insufficient, as shown by the counter-example x0 = 3 (p = 5 divides 10, 5 ≡ 1 (mod 4), yet 5 < 6 and the true minimal residue is 2).

minor comments (2)

Clarify the precise statement of the main theorem (e.g., whether the conditions are phrased in terms of the prime factors of x0² + 1 or in some other arithmetic form).
Add explicit cross-references between the historical narrative and the new arithmetic characterization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a key omission in our characterization of Størmer numbers. We address the major comment below and will revise the manuscript to incorporate the suggested clarification.

read point-by-point responses

Referee: The central claim (abstract and main theorem) asserts necessary and sufficient conditions for x0 to be the minimal positive solution of x² ≡ -1 (mod p). Sufficiency therefore requires that any prime p ≡ 1 (mod 4) dividing x0² + 1 satisfy p > 2x0. The manuscript must explicitly state and prove this size constraint; without it, the conditions are insufficient, as shown by the counter-example x0 = 3 (p = 5 divides 10, 5 ≡ 1 (mod 4), yet 5 < 6 and the true minimal residue is 2).

Authors: We agree that the referee has identified an important gap. For x0 to qualify as a Størmer number, it must be the smallest positive solution to the congruence, so any prime p ≡ 1 (mod 4) dividing x0² + 1 must indeed satisfy p > 2x0; otherwise a smaller positive residue exists. This size constraint was implicit in our reasoning but not stated explicitly in the theorem or proved. We will revise the main theorem to include the explicit requirement p > 2x0 and supply a short proof that this condition ensures minimality. The counter-example x0 = 3 with p = 5 is valid: since 5 < 6, x0 = 3 is not minimal (the true solution is 2), and we will note this distinction in the revised text. All examples appearing in the historical and computational sections of the paper already satisfy the constraint, so no further changes to those parts are needed. revision: yes

Circularity Check

0 steps flagged

No circularity: characterization of Størmer numbers rests on standard quadratic residue definitions and prime factorization

full rationale

The paper defines a Størmer number via the standard notion of the minimal positive solution x0 to x² ≡ -1 (mod p) for p prime ≡ 1 (mod 4), a definition drawn from classical proofs of Fermat's Two Squares Theorem. The central result supplies necessary and sufficient conditions on a natural number x0 for it to arise in this way for some such p. This amounts to checking existence of a suitable prime divisor p of x0² + 1 that satisfies the congruence class and minimality constraints; the argument proceeds from elementary properties of quadratic congruences and does not reduce any derived condition to a re-labeling of the input definition, a fitted parameter, or a self-citation chain. The derivation is therefore self-contained against external number-theoretic benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard number-theoretic background rather than new free parameters or invented entities. The result is a characterization within existing modular arithmetic.

axioms (1)

standard math Existence of a solution to x² ≡ -1 mod p when p is prime and p ≡ 1 mod 4
Invoked in the definition of Størmer numbers and their role in Fermat's Two Squares Theorem (abstract, first paragraph).

pith-pipeline@v0.9.0 · 5793 in / 1237 out tokens · 35812 ms · 2026-05-18T00:56:24.214908+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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A. J-H Yee,A peak into y-cruncher v0.6.1, http://www.numberworld.org/blogs/2012-3-9-a-peak-into- ycruncher-v0.6.1/, March 9, 2020. Department of Mathematics, Sid Richardson Building, 1410 S.4th Street, Waco, TX 76706. Email address:matthew kroesche@baylor.edu Department of Mathematics, Sid Richardson Building, 1410 S.4th Street, Waco, TX 76706. Email addr...

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G. E. Andrews,Number Theory,W. B. Saunders Company, Philadelphia, PA., 1971

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Egeland and W

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Everest and G

G. Everest and G. Harman, G,On primitive divisors of n 2+b, Number Theory and Polynomials, 142– 154, London Math. Soc. Lecture Note Ser., 352, Cambridge Univ. Press, Cambridge, 2008

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Hermite,Note au sujet de l’article pr´ ec´ edent, J

C. Hermite,Note au sujet de l’article pr´ ec´ edent, J. Math. Pures Appl. 13 (1848) 15

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Muir and W

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Serret,Sur un th´ eor` eme relatif aux nombres entiers, J

J.-A. Serret,Sur un th´ eor` eme relatif aux nombres entiers, J. Math. Pures Appl. 13 (1848), 12–14

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H. J. S. Smith,De Compositione Numerorum Primorum4λ+ 1Ex Duobus Quadratis,Crelle’s Journal L (1855), 91–92

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C. Størmer,Sur l‘application de la th´ eorie des nombres entiers complexes ´ a la solution en nombres rationnels de l‘` equationc1 arctangx1 +· · ·+c n arctangxn =kπ/4, Archiv for Mathematik og Naturvi- denskab, 19 (1896), no. 3, 1-96

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Størmer,Quelques th´ eor` emes sur l’´ equation de Pellx2 −Dy 2 =±1et leurs applications, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv

C. Størmer,Quelques th´ eor` emes sur l’´ equation de Pellx2 −Dy 2 =±1et leurs applications, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2) (1897)

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Størmer,Solution compl` ete en nombres entiers de l’´ equationm·arctan( 1 x) +n·arctan( 1 y ) =k π 4 , Bulletin de la S

C. Størmer,Solution compl` ete en nombres entiers de l’´ equationm·arctan( 1 x) +n·arctan( 1 y ) =k π 4 , Bulletin de la S. M. F., tome 27 (1899), pp. 160–170

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J. Ranous,StorageReview Lab Breaks Pi Calculation World Record with 202 Trillion Digits, http://www.storagereview.com/news/storagereview-lab-breaks-pi-calulation-world-record-with-over- 202-trillion-digits, June 28, 2024

work page 2024

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Todd,A Problem of Arc Tangent Relations, The American Mathematical Monthly, Vol

J. Todd,A Problem of Arc Tangent Relations, The American Mathematical Monthly, Vol. 56, No. 8, (1949), 517-528

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J-H Yee,A peak into y-cruncher v0.6.1, http://www.numberworld.org/blogs/2012-3-9-a-peak-into- ycruncher-v0.6.1/, March 9, 2020

A. J-H Yee,A peak into y-cruncher v0.6.1, http://www.numberworld.org/blogs/2012-3-9-a-peak-into- ycruncher-v0.6.1/, March 9, 2020. Department of Mathematics, Sid Richardson Building, 1410 S.4th Street, Waco, TX 76706. Email address:matthew kroesche@baylor.edu Department of Mathematics, Sid Richardson Building, 1410 S.4th Street, Waco, TX 76706. Email addr...

work page 2012