Heegner Numbers

My diurnal age today is \( \textbf{28163} \), a prime number, as is \( \textbf{163} \), the number formed by its last three digits. The latter number is quite famous in higher mathematics but I always forget why so I got Gemini to remind me.

The special quality of 163 that places it firmly in the realm of higher mathematics is that it is the largest of the nine Heegner numbers.

In algebraic number theory, a Heegner number is a square-free positive integer $d$ such that the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$ has class number $1$. Equivalently, this means the ring of integers of this field exhibits unique factorization. The complete set of Heegner numbers (cataloged as OEIS sequence A003173) is $1, 2, 3, 7, 11, 19, 43, 67$, and $163$.

The fact that $163$ ends this sequence leads to two profound and famous mathematical consequences: Ramanujan's constant and the deep structure behind Euler's prime-generating polynomial.

Ramanujan's Constant and Almost-Integers

Because 163 is the largest Heegner number, it produces an astonishing "almost-integer" when placed in the exponential function $e^{\pi \sqrt{d}}$. Specifically:

$$e^{\pi \sqrt{163}} \approx 262537412640768743.99999999999925007...$$

This is widely known as Ramanujan's constant. The reason this number is so staggeringly close to an integer is not a coincidence, but a direct result of the properties of the modular $j$-function.

For any Heegner number $d$, the $j$-invariant of the associated elliptic curve is an exact integer. For $d=163$, the $j$-invariant is exactly $-(640320)^3$.

The Fourier series expansion of the $j$-function in terms of $q = e^{2\pi i \tau}$ is:

$$j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2 + \dots$$

When we evaluate this at $\tau = \frac{1 + \sqrt{-163}}{2}$, $q$ becomes $-e^{-\pi \sqrt{163}}$. Because $163$ is very large, the higher-order terms ($196884q$, etc.) become infinitesimally small. By equating the exact integer $j$-invariant with the first two terms of the expansion, we get:

$$-(640320)^3 \approx -e^{\pi \sqrt{163}} + 744$$

Rearranging this yields:

$$e^{\pi \sqrt{163}} \approx (640320)^3 + 744 = 262537412640768744$$

Euler's Prime-Generating Polynomial

In 1772, Leonhard Euler published the famous quadratic polynomial $n^2 + n + 41$, which yields prime numbers for every integer $n$ from $0$ to $39$.

The remarkable prime-generating efficiency of this polynomial is directly tied to the field $\mathbb{Q}(\sqrt{-163})$. The discriminant of the polynomial $x^2 + x + 41$ is $1^2 - 4(1)(41) = -163$.

Rabinowitsch's theorem states that for $d \equiv 3 \pmod 4$, the polynomial $n^2 + n + \frac{d+1}{4}$ will output primes for all integers $n = 0, 1, \dots, \frac{d-3}{4}$ if and only if $d$ is a Heegner number. Because $163$ is the largest Heegner number, $\frac{163+1}{4} = 41$, making Euler's polynomial the absolute optimal case for this specific phenomenon. There is no other polynomial of this form that will generate a longer string of consecutive primes.

Photo of Kurt Heegner, this photo was taken at the
terrace of his mother's home in Berlin in the 1930s

I only partially understand the mathematics involved. The story behind Heegner, after whom the numbers are named, is as follows: 

The name attached to these numbers belongs to Kurt Heegner (1893–1965), and his story is one of the most famous and somewhat tragic examples of an "outsider" solving a major mathematical problem but dying before receiving recognition.

Here is the history of the man and why the numbers bear his name:

A Private Scholar, Not an Academic

Kurt Heegner was a German mathematician, physicist, and radio engineer. He was born and lived in Berlin for most of his life. Unlike the vast majority of names attached to profound mathematical discoveries, Heegner was not a professional academic. He held no university chair and spent much of his career working in radio engineering and patenting electronic circuits. Mathematics was a pursuit he undertook independently as a private scholar.

Gauss's Class Number Problem

The origin of the Heegner numbers dates back to 1801, when Carl Friedrich Gauss published his foundational text Disquisitiones Arithmeticae. Gauss conjectured that there were exactly nine imaginary quadratic fields with class number 1 (which correspond to the nine numbers $1, 2, 3, 7, 11, 19, 43, 67$, and $163$). For a century and a half, proving that there was no tenth number remained one of the outstanding unsolved problems in number theory.

The 1952 Proof and Rejection

In 1952, Heegner published a paper in a reputable mathematical journal claiming to have finally proven Gauss's conjecture. Heegner's approach was highly unorthodox for the time; he used complex modular functions (specifically building on the earlier work of Heinrich Weber) and Diophantine equations.

Unfortunately, the mathematical establishment completely dismissed his work. Reviewers found his paper dense, unconventional, and difficult to follow. It was widely concluded that his proof contained a fatal gap and was incomplete. Because Heegner was an unknown amateur without academic credentials to vouch for him, the mathematical community effectively ignored his publication, and the problem was still considered unsolved. Heegner died in 1965 without ever seeing his work accepted.

Posthumous Vindication

The story changed dramatically in 1966 and 1967. Two mathematicians, Alan Baker and Harold Stark, independently published universally accepted proofs of Gauss's conjecture using entirely different methods. Stark's proof, in particular, was celebrated as a major breakthrough.

However, in 1968, the British mathematician Bryan Birch (of the Birch and Swinnerton-Dyer conjecture) and Harold Stark himself decided to look back at Heegner's dismissed 1952 paper. Upon careful re-examination, they realized that Heegner’s proof was actually completely correct.

The supposed "gap" in Heegner's logic was not a flaw; he had simply omitted the details of a specific theorem from Weber's work, assuming the reader would know it. Once that theorem was explicitly applied, Heegner's proof was airtight. Stark and Birch brought this to the attention of the mathematical community, establishing that Heegner had successfully solved the problem 15 years before anyone else.

Legacy

To honor his undisputed priority and to rectify the community's failure to recognize a brilliant piece of mathematics during his lifetime, the nine numbers of Gauss's conjecture were formally named Heegner numbers. Furthermore, the points he used on elliptic curves to construct his proof are now known in modern arithmetic geometry as Heegner points, which remain a highly active and critical area of research in number theory today.