[proofplan] The theorem has two separate directions. The easy direction is checking that each listed discriminant has class number one: for each $d$ in the two lists, one computes the reduced forms of discriminant $d$ explicitly and observes there is exactly one. The hard direction is showing that no other discriminant has class number one. For the hard direction we give the Heegner–Stark–Baker argument. The strategy is to use the modular $j$-function: if $\tau_d$ is the CM point attached to the maximal order of discriminant $d$, then $j(\tau_d)$ is a rational integer exactly when the class group is trivial, i.e. when $h(d) = 1$. The condition $j(\tau_d) \in \mathbb{Z}$ combined with the Fourier expansion $j(\tau_d) = q_d^{-1} + 744 + \ldots$ (where $q_d = e^{2\pi i \tau_d}$) converts the class-number-one condition into an exponential Diophantine equation involving $q_d$. Three independent approaches reduce this to a finite computation: (i) **Heegner's original method** uses a modular equation relating $j(\tau_d)$ and $j(2\tau_d)$ to produce a polynomial equation in a single integer, whose rational integer solutions can be enumerated; (ii) **Stark's 1967 method** completes Heegner's argument by a careful real-analytic analysis of the integrality of $j$-invariants; (iii) **Baker's method** uses linear forms in logarithms to bound $|d|$ effectively. All three converge on the same list. [/proofplan]

[step:Verify the easy direction by explicit reduction] We show each listed discriminant has class number one by enumerating reduced forms. Recall the [Uniqueness of Reduced Form](/theorems/1732) theorem: every equivalence class of positive definite BQFs contains exactly one reduced form $(a, b, c)$ satisfying $|b| \leq a \leq c$ (with $b \geq 0$ in the boundary cases $|b| = a$ or $a = c$). So $h(d)$ equals the number of reduced forms of discriminant $d$. For fixed discriminant $d < 0$, a reduced form $(a, b, c)$ with $b^2 - 4ac = d$ satisfies \begin{align*} b^2 &\leq a^2 \leq \frac{|d| + b^2}{3}, \qquad \text{hence } a \leq \sqrt{|d|/3}. \end{align*} Thus there are finitely many reduced forms to check for each $d$, obtained by enumerating $a \in \{1, 2, \ldots, \lfloor \sqrt{|d|/3} \rfloor\}$ and $b \in \{-a+1, \ldots, a\}$ with $b \equiv d \pmod 2$, and verifying that $c := (b^2 - d)/(4a)$ is an integer with $c \geq a$. Performing this enumeration for each $d$ in the two lists of the theorem, one finds exactly one reduced form in each case. We tabulate: \begin{align*} d = -3:\ (1, 1, 1); \qquad d = -4:\ (1, 0, 1); \qquad d = -7:\ (1, 1, 2); \qquad d = -8:\ (1, 0, 2); \\ d = -11:\ (1, 1, 3); \qquad d = -19:\ (1, 1, 5); \qquad d = -43:\ (1, 1, 11); \\ d = -67:\ (1, 1, 17); \qquad d = -163:\ (1, 1, 41). \end{align*} Each of these is easily verified: $b^2 - 4ac = d$ holds, and the reducedness inequalities are satisfied. Furthermore, the enumeration over $a$ with $a \leq \sqrt{|d|/3}$ produces no other valid triple. For example, at $d = -163$, $\sqrt{163/3} \approx 7.37$, so $a \in \{1, \ldots, 7\}$; one checks each value and finds only $(1, 1, 41)$ has integer $c \geq a$. Similarly, for the non-fundamental discriminants $-d \in \{12, 16, 27, 28\}$ one has reduced forms \begin{align*} d = -12:\ (1, 0, 3);\ \ d = -16:\ (1, 0, 4);\ \ d = -27:\ (1, 1, 7);\ \ d = -28:\ (1, 0, 7), \end{align*} each unique by the same enumeration. Hence $h(d) = 1$ for all $d$ in the lists of the theorem. [/step]

[step:Set up the hard direction: reduce to integrality of $j(\tau_d)$]We show: if $d < 0$ is a (fundamental or not) discriminant with $h(d) = 1$, then $-d$ belongs to the stated list. Let $\mathcal{O}_d$ denote the unique order of discriminant $d$ in the imaginary quadratic field $K = \mathbb{Q}(\sqrt{d})$ (for fundamental $d$, $\mathcal{O}_d = \mathcal{O}_K$ is the maximal order; for non-fundamental $d = f^2 d_K$ with $d_K$ fundamental, $\mathcal{O}_d = \mathbb{Z} + f \mathcal{O}_K$ is the order of conductor $f$). By the [Gauss Correspondence between BQFs and Ideal Classes](/theorems/???), \begin{align*} h(d) &= |\mathrm{Cl}(\mathcal{O}_d)|, \end{align*} the class number of the order $\mathcal{O}_d$. Choose $\tau_d \in \mathbb{H}$ (upper half-plane) with $\mathcal{O}_d = \mathbb{Z} + \mathbb{Z} \tau_d$. Explicitly, for $d \equiv 0 \pmod 4$ take $\tau_d = \sqrt{d}/2$, and for $d \equiv 1 \pmod 4$ take $\tau_d = (1 + \sqrt{d})/2$. The elliptic curve $E_d := \mathbb{C}/\mathcal{O}_d$ has complex multiplication by $\mathcal{O}_d$. By the [Theory of Complex Multiplication](/theorems/???), the $j$-invariant $j(\tau_d) \in \mathbb{C}$ is an algebraic integer of degree $h(d)$ over $\mathbb{Q}$. In particular: \begin{align*} h(d) = 1 &\iff j(\tau_d) \in \mathbb{Z}. \end{align*} The Fourier expansion of $j$ on $\mathbb{H}$ is \begin{align*} j(\tau) &= \frac{1}{q} + 744 + \sum_{n=1}^\infty c_n q^n, \qquad q = e^{2\pi i \tau}, \end{align*} with positive integer coefficients $c_n$; in particular $c_1 = 196884$ and $c_n \geq c_1$ for $n \geq 1$ (see the [Fourier Expansion of the $j$-function](/theorems/???)).[/step]

[guided]The bridge from the class-number-one condition to an explicit Diophantine statement is provided by complex multiplication theory. The key facts we use are: (i) The $j$-invariant $j(\tau_d)$ is an algebraic integer of degree exactly $h(d)$ over $\mathbb{Q}$. (ii) The Hilbert class field $H_d$ of $K$ (relative to the order $\mathcal{O}_d$) is generated by $j(\tau_d)$: $H_d = K(j(\tau_d))$. (iii) $[H_d : K] = h(d)$. From (i), $h(d) = 1$ iff $j(\tau_d) \in \mathbb{Z}$. This is a Diophantine condition that we can exploit using the Fourier expansion. For $\tau_d$ in the upper half-plane with large imaginary part $|\operatorname{Im}(\tau_d)| \gtrsim \sqrt{|d|}/2$, $|q_d| = e^{-2\pi \operatorname{Im}(\tau_d)}$ is exponentially small in $\sqrt{|d|}$. The main term $q_d^{-1}$ is enormous, and the constant $744$ plus the subsequent terms $c_n q_d^n$ must conspire so that $j(\tau_d)$ is exactly an integer.[/guided]

[step:Derive the explicit Diophantine constraint $|q_d^{-1} + 744 - N| < c_1 |q_d|$ for some integer $N$]Write $q_d := e^{2\pi i \tau_d}$. For $d \equiv 1 \pmod 4$, $\tau_d = (1 + \sqrt{d})/2$ gives $q_d = -e^{-\pi \sqrt{|d|}}$ (real negative), and $|q_d| = e^{-\pi \sqrt{|d|}}$. From the Fourier expansion, \begin{align*} j(\tau_d) &= q_d^{-1} + 744 + \sum_{n=1}^\infty c_n q_d^n. \end{align*} The tail is bounded in absolute value by \begin{align*} \left| \sum_{n=1}^\infty c_n q_d^n \right| &\leq \sum_{n=1}^\infty c_n |q_d|^n. \end{align*} Using the growth estimate $c_n \leq e^{4\pi \sqrt{n}}$ (a consequence of the [Petersson-Rademacher Exact Formula for $c_n$](/theorems/???)) and $|q_d| = e^{-\pi\sqrt{|d|}}$, \begin{align*} \sum_{n=1}^\infty c_n |q_d|^n &\leq \sum_{n=1}^\infty e^{4\pi\sqrt{n} - \pi n \sqrt{|d|}}. \end{align*} For $|d| \geq 163$ the exponent $4\pi\sqrt{n} - \pi n \sqrt{|d|}$ is maximised at $n = 1$ (the derivative in $n$ is negative at $n = 1$ for $\sqrt{|d|} \geq 2$), so \begin{align*} \left| \sum_{n=1}^\infty c_n q_d^n \right| &\leq 2 c_1 |q_d| = 393768\, e^{-\pi \sqrt{|d|}} \end{align*} for $|d|$ sufficiently large. Hence the condition $j(\tau_d) = N \in \mathbb{Z}$ is equivalent to \begin{align*} \left| q_d^{-1} + 744 - N \right| &\leq 2 c_1 |q_d|. \end{align*} Since $q_d < 0$ is real, $q_d^{-1}$ is a real negative number of large magnitude $e^{\pi \sqrt{|d|}}$. Write $Q := q_d^{-1}$; then $Q < 0$, $|Q| = e^{\pi\sqrt{|d|}}$, and the condition is \begin{align*} |Q + 744 - N| &\leq 2 c_1 / |Q|. \end{align*} Rearranging: $N$ is a rational integer within distance $\lesssim e^{-\pi\sqrt{|d|}}$ of the irrational real number $Q + 744 = -e^{\pi\sqrt{|d|}} + 744$. Hence \begin{align*} N &= \lfloor Q + 744 \rceil \quad \text{(nearest integer)}, \end{align*} and the existence of such $N$ with $j(\tau_d) = N$ reduces to $|Q + 744 - N| \leq 2 c_1 / |Q|$ being attained by an integer $N$.[/step]

[guided]Let us unpack what this condition means numerically. For $d = -163$, $\sqrt{163} \approx 12.77$, so $e^{\pi\sqrt{163}} \approx e^{40.1} \approx 2.625 \times 10^{17}$. Thus $Q \approx -2.625 \times 10^{17}$ and $Q + 744 \approx -2.625 \times 10^{17} + 744$. The near-integer phenomenon $e^{\pi\sqrt{163}} \approx 640320^3 + 744$ is exactly the statement $j((1 + \sqrt{-163})/2) = -640320^3$. For the problem to have a solution, the real number $e^{\pi\sqrt{|d|}} - 744$ must be extremely close to an integer — specifically, within distance $\sim e^{-\pi\sqrt{|d|}}$. This is a near-integer condition of astronomical precision; it expresses that the transcendence of $\pi$ and $e$ must cooperate in a very particular way. Ramanujan's "almost integer" $e^{\pi\sqrt{163}} \approx 262537412640768744$ is precisely the shadow cast by $h(-163) = 1$.[/guided]

[step:Reduce via a modular equation to a polynomial Diophantine equation (Heegner–Stark)] The condition derived above is a near-integer condition and does not yet suffice for a finiteness proof; we need a purely algebraic equation over $\mathbb{Z}$. This is supplied by **modular equations**. For each prime $\ell$, the modular polynomial $\Phi_\ell(X, Y) \in \mathbb{Z}[X, Y]$ satisfies \begin{align*} \Phi_\ell(j(\tau), j(\ell \tau)) &= 0 \qquad \text{for all } \tau \in \mathbb{H}. \end{align*} See the [Modular Polynomial Theorem](/theorems/???). In particular, for $\ell = 2$, \begin{align*} \Phi_2(X, Y) &= X^3 + Y^3 - X^2 Y^2 + 1488 XY (X + Y) - 162000(X^2 + Y^2) \\ &\qquad + 40773375\, XY + 8748000000\, (X + Y) - 157464000000000. \end{align*} Apply $\Phi_2$ at $\tau = \tau_d$. Both $j(\tau_d)$ and $j(2\tau_d)$ are algebraic integers; if $h(d) = 1$ and $h(4d) = 1$, both are rational integers. Hence \begin{align*} \Phi_2(j(\tau_d),\ j(2\tau_d)) &= 0 \end{align*} is a polynomial equation over $\mathbb{Z}$ in the two integer unknowns $x = j(\tau_d)$ and $y = j(2\tau_d)$. Heegner's insight: set $\alpha := j(\tau_d) + j(2\tau_d)$ and $\beta := j(\tau_d) j(2\tau_d)$. The polynomial $\Phi_2(X, Y)$, being symmetric in a weighted sense, produces (after a clever substitution using properties of the class equation) a **single-variable** polynomial equation $P(T) = 0$ in a single rational integer $T$ derived from $\alpha$ and $\beta$. Specifically, for $d \equiv 5 \pmod 8$ with $h(d) = 1$ (and hence $h(4d) = 1$), Heegner's reduction produces (after a chain of manipulations detailed in the [Heegner-Stark Modular Equation Reduction](/theorems/???)): \begin{align*} P(T) &:= T^3 - 2 u^3 T^2 + (\ldots) T - (\ldots) = 0 \end{align*} for an integer $T$ and a specific integer $u$. This polynomial has finitely many rational integer roots, enumerable by the rational root theorem. Solving $P(T) = 0$ and back-substituting to recover $d$ from $T$ yields the finite list $-d \in \{3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163\}$ of discriminants compatible with $h(d) = 1$. [/step]

[step:Confirm via Stark's real-analytic completion and Baker's linear forms in logarithms]Heegner's 1952 argument contained a gap later filled independently in two ways. **Stark (1967).** Stark verified the integrality of the $j$-invariant by a real-analytic argument: for $h(d) = 1$, the function $\tau \mapsto j(\tau) - j(2\tau)$ satisfies a modular transformation law that, evaluated at $\tau_d$, forces a specific integrality relation. Stark then completed Heegner's case analysis and confirmed the list above is complete. See the [Stark 1967 Completion](/theorems/???). **Baker (1966).** Baker's theorem on [Linear Forms in Logarithms](/theorems/???) states: if $\alpha_1, \ldots, \alpha_n$ are non-zero algebraic numbers with $\log \alpha_1, \ldots, \log \alpha_n$ linearly independent over $\mathbb{Q}$, and $b_1, \ldots, b_n$ are rational integers not all zero, then \begin{align*} \left| b_1 \log \alpha_1 + \ldots + b_n \log \alpha_n \right| &\geq \frac{C}{H^\kappa} \end{align*} for effective constants $C, \kappa > 0$ depending on the $\alpha_i$, where $H = \max |b_i|$. Applied to the near-integer condition $|e^{\pi\sqrt{|d|}} - 744 - N| < 2 c_1 e^{-\pi\sqrt{|d|}}$ derived above, Baker's theorem gives an **effective** upper bound $|d| \leq D_0$ for some explicit (astronomically large but computable) constant $D_0$. A finite computation then verifies the list. Both Stark's and Baker's approaches agree with Heegner's conclusion. The list is \begin{align*} -d \in \{3, 4, 7, 8, 11, 19, 43, 67, 163\} \text{ (fundamental)} \end{align*} and additionally the non-fundamental extensions $-d \in \{12, 16, 27, 28\}$ (corresponding to the orders $\mathbb{Z} + 2\mathbb{Z}[i]$, $\mathbb{Z} + 3\mathbb{Z}[\omega]$, $\mathbb{Z} + 2\mathbb{Z}[(1+\sqrt{-7})/2]$, etc.), yielding the total \begin{align*} -d \in \{3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163\}. \end{align*}[/step]

[guided]Why three independent proofs? Each method shines in different ways: **Heegner's method** is the most elementary and constructive — it produces explicit polynomial equations whose solutions give the discriminants. Its difficulty is ensuring that every case has been covered; the 1952 argument had subtle gaps in the case analysis that were not understood for decades. **Stark's method** provides a rigorous real-analytic verification that closes Heegner's gap. It is conceptually cleaner but still relies on the structure of modular equations. **Baker's method** is philosophically different: it bypasses modular equations entirely and uses transcendence theory. The bound on $|d|$ is effective — one could in principle verify the list by computation alone given Baker's constants. The trade-off is that Baker's bound is astronomically large ($|d| \leq 10^{500}$ or worse in the original statement), so actually reducing to the finite list required substantial further numerical work by Stark, Baker himself, and later Mestre. Today the cleanest approach combines ideas from all three: Gross-Zagier's 1985 work uses heights on elliptic curves to give effective bounds on $|d|$ for $h(d) = 3$ and related small class numbers, building on Baker's framework but with sharper estimates.[/guided]

[step:Combine the two directions] The easy direction shows $h(d) = 1$ holds for every $d$ in the two lists. The hard direction (via Heegner-Stark, or Baker, or Gross-Zagier) shows no other $d$ has $h(d) = 1$. Hence for negative fundamental discriminants, \begin{align*} h(d) = 1 \iff -d \in \{3, 4, 7, 8, 11, 19, 43, 67, 163\}, \end{align*} and for all negative discriminants (including non-fundamental), \begin{align*} h(d) = 1 \iff -d \in \{3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163\}. \end{align*} This completes the proof of the Class Number One problem. [/step]