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Carl Størmer and his Numbers

This paper establishes necessary and sufficient conditions for a natural number to be a Størmer number—defined as the smallest least residue solution to $x^2 \equiv -1 \bmod p$ for primes $p \equiv 1 \bmod 4$ —while also exploring Carl Størmer's historical work connecting these numbers to Gregory numbers and high-precision approximations of $\pi$ .

Original authors: Matthew Kroesche, Lance L. Littlejohn, Graeme Reinhart

Published 2026-04-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthew Kroesche, Lance L. Littlejohn, Graeme Reinhart

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about a special club of numbers. This paper is about that club, the rules for joining, and how these numbers help us solve a very old puzzle about the number $\pi$ (pi).

Here is the story of Carl Størmer's Numbers, broken down into simple parts.

1. The Club: "Størmer Numbers"

First, let's talk about a famous math rule called Fermat's Two Squares Theorem. It says that if you have a prime number (a number only divisible by 1 and itself) that is one more than a multiple of 4 (like 5, 13, 17, 29), you can always split it into two perfect squares added together.

Example: $13 = 2^2 + 3^2$ ( $4 + 9$ ).

To find those two squares ($2$ and $3$), mathematicians usually have to solve a tricky riddle first: Find a number $x$ such that when you square it and divide by the prime, the remainder is $-1$.

Example: For the prime 13, if you square 5, you get 25. $25$ divided by $13$ leaves a remainder of $-1$ (or $12$). So, 5 is the key.

The authors call this key number a Størmer Number. It's like the "password" that unlocks the door to splitting the prime number into two squares.

2. The Big Discovery: Who Gets the Password?

For a long time, people knew how to find the password if they already had the prime number. But what if you just have a random number, like 15? Is 15 a password for some prime? Or is it a fake?

The authors of this paper figured out the exact rule to tell if a number is a real Størmer Number or not.

The Rule (The "Bouncer" at the Door):
Take your number (let's call it $x$ ). Square it and add 1 ( $x^2 + 1$ ). Now, look at all the prime numbers that multiply together to make that result.

Find the largest prime in that group.
The Rule: Your number $x$ is a Størmer Number if and only if that largest prime is bigger than $2x + 1$ .

If the largest prime is too small, $x$ is just a regular number, not a Størmer Number. If it's big enough, $x$ is the unique password for that specific large prime.

Analogy: Imagine you have a key ( $x$ ). You check the lock it fits ( $x^2+1$ ). If the biggest bolt in the lock is huge (bigger than twice your key size), then you have a real Størmer key. If the bolt is small, you're holding a fake key.

3. How Common Are They?

The authors looked at the first million numbers and found that about 70% of them are Størmer Numbers.
They suspect that as you go to infinity, this percentage settles exactly at $\ln(2)$ (which is about 0.693, or 69.3%). They didn't prove this with 100% certainty (it's still a "heuristic" or educated guess), but the numbers strongly suggest it's true.

4. The Real-World Hero: Carl Størmer

The paper also introduces the man behind the numbers: Carl Størmer (1874–1957).

Who was he? A brilliant Norwegian mathematician and astronomer.
What else did he do? He was famous for studying the Aurora Borealis (Northern Lights). He figured out how charged particles from the sun dance in Earth's magnetic field. He was so respected that a crater on the Moon is named after him!
The Fun Fact: He was also a secret photographer. He carried a tiny spy camera in his jacket and took candid photos of people on the streets of Oslo. Later, these photos became a famous art exhibition.

5. The Grand Finale: Calculating Pi ( $\pi$ )

Why do we care about these numbers? Because they help us calculate $\pi$ (the ratio of a circle's circumference to its diameter).

In the old days, before computers, calculating $\pi$ was like trying to fill a swimming pool with a teaspoon. You needed a very efficient way to do it. Mathematicians use formulas involving arctangents (a type of angle calculation) to get the digits of $\pi$ .

Størmer discovered a special connection:

If you have a number that is NOT a Størmer Number, you can break its arctangent formula down into a sum of arctangents of Størmer Numbers.
If a number IS a Størmer Number, it is "irreducible"—you can't break it down further. It's a fundamental building block.

The "Super Formula":
In 1896, Størmer found a complex formula to calculate $\pi$ .

In 2002, a team led by Yasumasa Kanada used Størmer's formula to calculate 1.24 trillion digits of $\pi$ .
This was a world record at the time.

So, a number theory concept from the 19th century, defined by a "bouncer" rule about prime factors, was the secret weapon used to break the world record for calculating the digits of $\pi$ in the 21st century.

Summary

Størmer Numbers are special "passwords" that help split certain prime numbers into two squares.
The paper gives a simple test to see if any number is a Størmer Number: Check if the largest prime factor of ( $x^2+1$ ) is bigger than $2x+1$ .
About 69% of all numbers are Størmer Numbers.
These numbers were crucial for Carl Størmer to create formulas that allowed computers to calculate trillions of digits of $\pi$ .
Carl Størmer was also a moon-crater-naming astronomer and a street-photography spy.

1. Problem Statement

The paper addresses the characterization of Størmer numbers, a specific class of integers arising from the study of Fermat's Two Squares Theorem and the approximation of $\pi$ .

Context: For a prime $p \equiv 1 \pmod 4$ , Fermat's Two Squares Theorem guarantees a unique representation $p = a^2 + b^2$ . Most constructive proofs of this theorem rely on finding the smallest positive integer $x_0$ such that $x_0^2 \equiv -1 \pmod p$ and $1 < x_0 \le (p-1)/2$ .
The Gap: While the existence of such an $x_0$ is known, there was no simple, necessary, and sufficient condition to determine if an arbitrary integer $x_0 \in \mathbb{N}$ is the Størmer number for some prime $p$ . Furthermore, the statistical distribution (natural density) of these numbers was a subject of conjecture but lacked a rigorous proof.
Historical Motivation: The study connects to Carl Størmer's work on identities expressing $\pi$ as linear combinations of arctangent values (Gregory numbers), which were crucial for high-precision $\pi$ calculations before the computer age.

2. Methodology

The authors employ a combination of number theory, algebraic number theory (specifically Gaussian integers), and probabilistic heuristics.

Continuants and Euclidean Algorithm: The paper reviews Smith's method, which uses the Euclidean algorithm and continuants (determinants of tridiagonal matrices) to link the solution of $x^2 \equiv -1 \pmod p$ to the decomposition $p = a^2 + b^2$ .
Prime Factorization Analysis: The core methodology involves analyzing the prime factorization of $x_0^2 + 1$ . The authors investigate the relationship between $x_0$ and the largest prime factor of $x_0^2 + 1$ .
Gaussian Integers: To address the connection to Gregory numbers, the authors utilize the ring of Gaussian integers $\mathbb{Z}[i]$ . They map the problem of decomposing $\arctan(1/n)$ to the factorization of Gaussian integers $n+i$ and the properties of their norms.
Probabilistic Heuristics: For the density conjecture, the authors construct a heuristic argument based on the distribution of primes in arithmetic progressions and Mertens' theorems.

3. Key Contributions and Results

A. Characterization of Størmer Numbers (Theorem 2.3)

The paper establishes a definitive necessary and sufficient condition for an integer $x_0$ to be a Størmer number.

Definition: Let $x_0^2 + 1 = 2^r \cdot p_1^{e_1} \cdots p_m^{e_m}$ be the prime factorization, where $p_1 < p_2 < \dots < p_m$ .
Theorem: $x_0$ is the Størmer number for a prime $p$ if and only if $2x_0 + 1 \le p_m$ (where $p_m$ is the largest prime factor of $x_0^2 + 1$ ).
Implication: If the condition holds, the associated prime is unique and equals $p = p_m$ . If $2x_0 + 1 > p_m$ , then $x_0$ is not a Størmer number for any prime.
Injectivity: The authors prove that the Størmer function $S(p)$ is injective; a specific $x_0$ corresponds to at most one prime $p$ .

B. Natural Density Conjecture (Section 3)

Conjecture: The paper discusses the conjecture by Everest and Harman (2008) that the natural density of Størmer numbers is $\ln(2) \approx 0.693$ .
Heuristic Proof: The authors provide a probabilistic argument suggesting this density. They estimate the probability that a random $x_0$ satisfies the condition $2x_0 + 1 \le p \le x_0^2 + 1$ for some prime $p$ . Using the Prime Number Theorem for arithmetic progressions and Mertens' estimates, they derive the limit:
$\lim_{x_0 \to \infty} \sum_{2x_0 \le p \le x_0^2} \frac{1}{p} \sim \ln(\ln(x_0^2)) - \ln(\ln(2x_0)) \sim \ln(2)$
Note: The authors explicitly state that a rigorous proof of this density remains elusive.

C. Connection to Gregory Numbers and $\pi$ (Section 5)

Størmer's Theorem: The paper formalizes Størmer's result that if $\gcd(a, b) = 1$ , then $\arctan(b/a)$ can be uniquely expressed as a finite linear combination of $\arctan(1/n)$ where each $n$ is a Størmer number.
Mechanism: This is proven via unique factorization in $\mathbb{Z}[i]$ . The decomposition of $a+bi$ into Gaussian primes corresponds to decomposing the angle $\arctan(b/a)$ into angles associated with Størmer numbers.
Irreducibility: The paper connects Størmer numbers to "irreducible" Gregory numbers (in the sense of Todd), proving that $n$ is a Størmer number if and only if $\arctan(1/n)$ cannot be written as a sum of other $\arctan(1/k)$ terms.
Historical Application: The paper highlights how Størmer's 1896 identity ( $t_1 = 44t_{57} + 7t_{239} - 12t_{682} + 24t_{12943}$ ) was used by Yasumasa Kanada in 2002 to calculate 1.24 trillion digits of $\pi$ .

4. Significance

Theoretical Clarity: The paper resolves the ambiguity regarding the identification of Størmer numbers, providing a computationally efficient test (Theorem 2.3) that avoids the need to search for the specific prime $p$ associated with an $x_0$ .
Bridging Fields: It effectively bridges classical number theory (Fermat's theorem, quadratic congruences) with the history of computational mathematics (approximations of $\pi$ ).
Historical Context: The paper serves as a comprehensive tribute to Carl Størmer, detailing his dual legacy in number theory and astrophysics (specifically the study of auroras and charged particles in magnetic fields).
Algorithmic Utility: The characterization allows for the generation of Størmer numbers and the construction of new arctangent identities for $\pi$ , which, while less critical for modern supercomputers, remains mathematically elegant and historically significant.

In summary, the paper provides a rigorous characterization of Størmer numbers, offers a compelling heuristic for their density, and elucidates their fundamental role in the algebraic structure of arctangent identities used to approximate $\pi$ .