Class Number Problem

Infinitely Many Real Quadratic Fields of Class Number One

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A real quadratic field is $K={\mathbb{Q}}(\sqrt d)$ for a squarefree integer $d>1$. Its ring of integers is $\mathcal O_K$, and its ideal class group is the group of nonzero fractional ideals modulo principal ideals. Unique factorization of elements in $\mathcal O_K$ is equivalent to this group having order $h_K=1$. Distinct squarefree $d$ give the fields being counted. [S1] [S2]

Statement recorded in the supplied source.

Are there infinitely many real quadratic number fields with unique factorization? [S1]

Short English statement

Are there infinitely many real quadratic number fields whose rings of integers have unique factorization?

Research attempt

Scope and method.

The problem concerns ordinary ideal class number one for real quadratic fields, not imaginary fields, narrow class groups, or nonmaximal orders. For squarefree $d>1$, the discriminant is $D=d$ when $d\equiv1\pmod4$ and $D=4d$ otherwise. The ring is correspondingly $\mathbb Z[(1+\sqrt d)/2]$ or $\mathbb Z[\sqrt d]$. The Minkowski method stated in the author notes [E1], inspected 1 October 2026, gives an exact finite certificate for any selected field. The attempt studies what would make such certificates uniform in a family.

The finite certificate and its quantitative limitation.

Minkowski's ideal-class theorem in degree two with two real embeddings says that every ideal class contains an integral ideal of norm at most \[ \frac{\sqrt D}{2}. \] This standard theorem is used as an input. Therefore it suffices to show that every prime ideal with norm at most that bound is principal: factor any small-norm representative into prime ideals and multiply their principal generators. The conclusion is about ideal classes, not equality of the ideals themselves.

If $D<16$, the bound is less than two, so the only possible representative norm is one; its ideal is the whole ring. This proves class number one for $d=2,3,5,13$. It produces four fields, not an infinite family. As $D$ increases the bound grows, and additional small prime ideals must be dealt with. Finiteness for each fixed field does not mean a uniform finite list of prime ideals works for every field.

Explicit certificates beyond the automatic range.

For $d=6$, the bound is $\sqrt{24}/2<3$, so only prime ideals of norm two can matter. The prime two ramifies, giving one such prime ideal. The element $2+\sqrt6$ has norm $-2$, so its principal ideal has norm two and is that unique prime ideal. Thus $h_{\mathbb Q(\sqrt6)}=1$. For $d=7$, the same argument uses $3+\sqrt7$ of norm two and the bound $\sqrt{28}/2<3$, again proving class number one.

For $d=17$, the ring contains $(1+\sqrt{17})/2$ and the bound is $\sqrt{17}/2<3$. The prime two splits. The element $(5+\sqrt{17})/2$ has norm two, so one of the two norm-two prime ideals is principal, and conjugation makes the other principal too. Thus this field also has class number one. These are complete small certificates, not just examples of elements with small norm whose ideal classes have not been linked to the required generators.

A small fundamental-looking unit does not force class number one.

The family $d=t^2+1$ has the unit $t+\sqrt d$ of norm $-1$ whenever $d$ is squarefree. It is tempting to view that explicit unit as a certificate of unique factorization, but units and ideal classes are different invariants. For a concrete obstruction take $d=26$ and $t=5$. The ideal $\mathfrak p=(2,\sqrt{26})$ has norm two, since the quotient ring is $\mathbb F_2$. If it were principal, a generator would have absolute norm two. Since the ring is $\mathbb Z[\sqrt{26}]$, that would require integers $x,y$ with \[ x^2-26y^2=2\quad\hbox{or}\quad-2. \] Modulo thirteen, this would make $x^2$ equal to two or eleven. The square residues are $0,1,3,4,9,10,12$, so neither is possible. Hence $\mathfrak p$ is nonprincipal and the class number exceeds one, despite the explicit norm-minus-one unit $5+\sqrt{26}$.

This counterexample blocks a proposed proof mechanism, not the original infinitude conjecture. It shows that controlling a Pell solution alone does not automatically principalize the small ideal classes required by Minkowski's theorem. The argument uses exact modular arithmetic and does not require a numerical class-number calculation for $d=26$.

The family-level task left by this approach.

To obtain infinitely many fields via finite certificates, one needs infinitely many distinct squarefree $d$ and a uniform method making all prime ideals up to $\sqrt D/2$ principal for each of them. Producing solutions to a selected norm equation handles at most selected ideal classes; a growing list of other ideals remains. Conversely, failure of a chosen family does not prove finiteness across all real quadratic fields.

The ring-of-integers convention is also indispensable. When $d\equiv1\pmod4$, testing only equations $x^2-dy^2=\pm p$ in integer $x,y$ can miss generators with half-integer coefficients in the actual ring. The $d=17$ certificate illustrates this concretely. All assertions above use the maximal order specified in the question.

Verification and outcome.

The displayed norm identities and the complete list of residues modulo thirteen are directly checkable finite certificates; no external class-group program was run or claimed. The global theorem input is Minkowski's bound, explicitly identified above. The attempt proves several individual class-number-one cases and an obstruction to a plausible unit-based family argument, but supplies no infinite family. Outcome: UNRESOLVED. The gap is uniform principalization of all required ideal classes over infinitely many distinct fields.

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1271 (NT-064), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 List of number fields with class number one. https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one. Bibliographic information from the supplied record.

• E1 Keith Conrad, Class Group Calculations, author lecture notes, Section 1 and quadratic-field examples. https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgpex.pdf. Inspected 1 October 2026. Used for the standard Minkowski theorem; the explicit norm computations in this attempt are given in full.