For a squarefree integer d>1d>1, let h(d)h(d) be the ideal class number of the real quadratic field Q(d)\mathbb{Q}(\sqrt{d}). Gauss conjectured in 1801 that infinitely many real quadratic fields have class number one; the imaginary quadratic counterpart was completely settled in the 1960s, but the real case remains open. Issa and Sankari (2020) give sufficient criteria for h(d)>1h(d)>1 in certain parametric families — via solvability of Pell-type norm equations x2−dy2=mx^{2}-dy^{2}=m, principality of ideals in quadratic orders, and continued-fraction period congruences — and address only those families; the general problem remains open as documented by Yokoi (1988) and Biró (2022). Effective determinations of class-number-one fields exist only family by family (Mollin–Williams for the Shanks sequence, Louboutin–Mollin–Williams under the Riemann Hypothesis, Biró's resolutions of Yokoi's and Chowla's conjectures and its 2022 extension), and the Cohen–Lenstra heuristics predict that the odd part of the class group is trivial for about 75% of real quadratic fields, which — combined with genus theory — supports infinitude without proving it. The general problem — Gauss's infinitude conjecture and a complete effective characterization — remains open.

Background

A real quadratic field is K=Q(d)K=\mathbb{Q}(\sqrt{d}) for a squarefree integer d>1d>1; its ring of integers is Z[d]\mathbb{Z}[\sqrt{d}] when d≡2,3(mod4)d\equiv 2,3\pmod 4 and Z[(1+d)/2]\mathbb{Z}[(1+\sqrt{d})/2] when d≡1(mod4)d\equiv 1\pmod 4. The ideal class group of this ring measures the failure of unique factorization, and its order h(d)h(d), the class number, equals one exactly when every ideal is principal. Unlike imaginary quadratic fields, real quadratic fields have infinitely many units; a generator εD>1\varepsilon_D>1 of the unit group (for discriminant DD) is the fundamental unit, computable from the periodic continued fraction of d\sqrt{d}, whose period length also governs the arithmetic of the field. Dirichlet's class number formula links these invariants: h(D)log⁡εD=D2 L(1,χD)h(D)\log\varepsilon_D=\tfrac{\sqrt{D}}{2}\,L(1,\chi_D), where L(1,χD)L(1,\chi_D) is the Dirichlet LL-function of the quadratic character χD\chi_D. In his Disquisitiones Arithmeticae (1801), Gauss conjectured that infinitely many real quadratic fields have class number one. The imaginary analogue — determining all imaginary quadratic fields of class number one, exactly nine of them — was completed in the 1960s, with Stark (1967) giving the final complete determination; there the formula contains no regulator term. In the real case the unbounded fundamental unit εD\varepsilon_D entangles h(D)h(D) with an object of uncontrolled size, and the best unconditional lower bounds for L(1,χD)L(1,\chi_D) (Siegel's theorem) are ineffective, which blocks uniform effective arguments, as recalled explicitly in Biró's 2022 Mathematika paper. A parallel heuristic strand — the Cohen–Lenstra heuristics, which model class groups of number fields as random finite abelian groups with the prime 2 excluded — predicts that the odd part of the class group of a real quadratic field is trivial (equivalently, the class number is a power of 2) for about 75.446% of such fields. This is not a positive-density statement about class number one: by genus theory, class number one forces the 2-rank of the class group to be zero, which forces the discriminant to have at most two prime factors (with a parity condition tied to the sign of the norm of the fundamental unit), a density-zero condition. Combined, the heuristics suggest infinitely many class-number-one fields forming a density-zero but unbounded set, supporting Gauss's conjecture without proving it. Progress has therefore proceeded family by family, largely for Richaud–Degert-type discriminants (those of the shape D=n2+rD=n^{2}+r with small rr, for which the continued fraction of d\sqrt{d} and the fundamental unit are explicit): Mollin–Williams (1991) solved the class number one problem for Shanks's sequence of discriminants, Louboutin–Mollin–Williams (1992) determined the class-number-one fields of a broad such family conditionally on the Riemann Hypothesis using quadratic residue covers, and Biró (2003) unconditionally settled Yokoi's conjecture (fields Q(n2+4)\mathbb{Q}(\sqrt{n^{2}+4})) and Chowla's conjecture (fields Q(4m2+1)\mathbb{Q}(\sqrt{4m^{2}+1})), extending these in 2022 to discriminants D=(b(1+bc)n+c)2+4(1+bc)nD=(b(1+bc)n+c)^{2}+4(1+bc)^{n} with bb divisible by a fixed integer. Within this tradition, Issa and Sankari (2020) derive sufficient criteria for h(d)>1h(d)>1 in further parametric families, using the solvability of the general Pell equation x2−dy2=mx^{2}-dy^{2}=m (a norm equation in the quadratic order), principality of index-aa ideals, and congruences on the continued-fraction period of d\sqrt{d}, while addressing only these special families, so that the general question remains untouched.

Problem Statement

Let d>1d>1 range over squarefree integers and let h(d)h(d) denote the ideal class number of the real quadratic field Q(d)\mathbb{Q}(\sqrt{d}). Resolve Gauss's class number one problem for real quadratic fields in its full generality, the general problem left open in the surrounding literature and not addressed by the family-wise criteria of Issa and Sankari (2020):

(i) Infinitude. Prove or disprove Gauss's conjecture that there are infinitely many squarefree dd with h(d)=1h(d)=1.

(ii) Effective determination. Give an unconditional and effective characterization of the complete set of squarefree integers dd with h(d)=1h(d)=1 — for instance an explicit structural description applying to all families of discriminants (a finite list if finiteness holds, or an effectively decidable parametric/structural criterion if infinitude holds; since h(d)h(d) is computable for each fixed dd, a bare per-dd class-number computation does not by itself constitute such a characterization; in the infinitude case the criterion must be a single predicate P(d)P(d), uniform in dd and given by an explicit condition on dd, such that P(d)  ⟺  h(d)=1P(d)\iff h(d)=1 and PP is not literally a per-dd evaluation of h(d)h(d)) — together with a proof of its correctness that does not rely on unproved hypotheses such as the Riemann Hypothesis, nor on ineffectively computed constants.

(ii) is the stronger of the two requirements and subsumes (i): an effective characterization of the complete set settles whether that set is finite or infinite, so a complete solution must include (ii), whereas a proof or disproof of (i) alone resolves only part (i).

Both parts target the single overarching objective left open by the family-by-family state of the art: the special-family criteria and determinations (Pell-solvability and continued-fraction criteria, Yokoi, Chowla, Shanks, Richaud–Degert-type families, and their extensions) cover only discriminants with explicitly computable fundamental units, and the general question — which real quadratic fields have class number one, and whether they are infinite in number — remains unanswered.

References

  1. Ahmad Issa and Hasan Sankari, "Some Criteria for Class Numbers to Be Non-One", Journal of Mathematics, vol. 2020 (2020), Article 5672983, pp. 1–5. DOI: 10.1155/2020/5672983, https://doi.org/10.1155/2020/5672983
  2. Carl Friedrich Gauss, Disquisitiones Arithmeticae (original 1801); English translation, Yale University Press. DOI: 10.12987/9780300194258, https://doi.org/10.12987/9780300194258
  3. Hideo Yokoi, "Class number one problem for real quadratic fields (the conjecture of Gauss)", Proceedings of the Japan Academy, Series A, Mathematical Sciences, vol. 64 (1988), no. 2, pp. 53–55. DOI: 10.3792/pjaa.64.53, https://doi.org/10.3792/pjaa.64.53
  4. H. M. Stark, "A complete determination of the complex quadratic fields of class-number one", Michigan Mathematical Journal, vol. 14 (1967), no. 1, pp. 1–27. DOI: 10.1307/mmj/1028999653, https://doi.org/10.1307/mmj/1028999653
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  6. S. Louboutin, R. A. Mollin, and H. C. Williams, "Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials and quadratic residue covers", Canadian Journal of Mathematics, vol. 44 (1992), no. 4, pp. 824–842. DOI: 10.4153/cjm-1992-049-0, https://doi.org/10.4153/cjm-1992-049-0
  7. András Biró, "Yokoi's conjecture", Acta Arithmetica, vol. 106 (2003), no. 1, pp. 85–104. DOI: 10.4064/aa106-1-6, https://doi.org/10.4064/aa106-1-6
  8. András Biró, "Chowla's conjecture", Acta Arithmetica, vol. 107 (2003), no. 2, pp. 179–194. DOI: 10.4064/aa107-2-5, https://doi.org/10.4064/aa107-2-5
  9. András Biró, "Class number one problem for a family of real quadratic fields", Mathematika, vol. 68 (2022), no. 4, pp. 1221–1232. DOI: 10.1112/mtk.12164, https://doi.org/10.1112/mtk.12164
  10. H. Cohen and H. W. Lenstra, Jr., "Heuristics on class groups of number fields", in Number Theory, Noordwijkerhout 1983, Lecture Notes in Mathematics, vol. 1068, Springer, 1984, pp. 33–62. DOI: 10.1007/bfb0099440, https://doi.org/10.1007/bfb0099440
  11. Ahmad Issa and Hasan Sankari, "A Necessary and Sufficient Condition for k=Q(4n2+1)k=\mathbb{Q}(\sqrt{4n^{2}+1}) to Have Class Number ω(n)+c\omega(n)+c", International Journal of Mathematics and Mathematical Sciences, vol. 2022 (2022), Article 6579960. DOI: 10.1155/2022/6579960, https://doi.org/10.1155/2022/6579960
  12. Ahmad Issa, "Criterion for Class Number of the Family Q(254m2+m)\mathbb{Q}(\sqrt{254m^{2}+m}) to be Greater Than or Equal to Three", Journal of Mathematics, vol. 2025 (2025), Article 4051906. DOI: 10.1155/jom/4051906, https://doi.org/10.1155/jom/4051906