Unconditional completion of the class number one problem for extended Richaud–Degert type real quadratic fields

A real quadratic field $\mathbb{Q}(\sqrt{d})$ is of extended Richaud–Degert (ERD) type when the squarefree radicand $d$ has the form $d=m^{2}+r$ with $r\mid 4m$ (classically with $-m<r\leq m$ or $r=\pm 4m/3$). Building on work of Louboutin, Yokoi, and Mollin–Williams, the class number one problem for this family was reduced to an explicit list of 43 fields together with at most one further, undetermined exceptional field; the unresolved single case is an artifact of Tatuzawa's ineffective lower bound for Dirichlet $L$-functions, and under the generalized Riemann hypothesis (GRH) at most one additional field can occur. The open problem is to settle, without assuming any unproved hypothesis, whether the known 43-field list is complete, and if not, to determine the exceptional field. Subfamilies have been settled unconditionally by Biró (Chowla's and Yokoi's conjectures), Byeon–Kim–Lee (Mollin's conjecture), Lee (ERD types not congruent to 5 modulo 8), Biró–Lapkova ($d=(an)^{2}+4a$), and Hoque–Kotyada ($d=a^{2}m^{2}+4ap$), but the full unconditional determination remains open as of 2026.

Background

For a number field $K$, the ideal class group $\mathrm{Cl}(K)$ is the finite abelian group measuring the failure of unique factorization of integers in the ring of algebraic integers of $K$; its order $h(K)$ is the class number, and $h(K)=1$ exactly when every ideal is principal. For the real quadratic fields $K=\mathbb{Q}(\sqrt{d})$ ($d>1$ squarefree), Gauss conjectured around 1801 that infinitely many have class number one; this remains one of the most famous open problems of algebraic number theory, with no essential progress toward an unconditional proof.

Since the class number formula ties $h(K)$ to the residue of the Dedekind zeta function at $s=1$, which involves the regulator (a logarithm of a fundamental unit), progress has concentrated on families where the fundamental unit — hence the regulator — is explicitly controllable. The oldest such family goes back to Richaud (1866) and Degert (1958): $d$ is of Richaud–Degert type if $d=m^{2}+r$ with $r\mid 4m$ and $-m<r\leq m$ (or $r=\pm 4m/3$); the extended Richaud–Degert (ERD) type relaxes the size restriction on $r$ (terminology varies slightly across authors). For these fields the continued fraction of $\sqrt{d}$ — and hence the fundamental unit — has a short parametric shape, giving $R_{K}<3\log(3d)$.

By the analytic class number formula together with Tatuzawa's theorem (an ineffective lower bound for $L(1,\chi)$, valid for all Dirichlet characters with at most one exceptional conductor), Louboutin, Yokoi, and Mollin–Williams (1990) determined the ERD fields of class number one with one possible exception: there are exactly the 43 fields with

$$d\in\{2,3,5,6,7,11,13,14,17,21,23,29,33,37,38,47,53,62,69,77,83,93,101,141,167,173,197,213,227,237,293,398,413,437,453,573,677,717,1077,1133,1253,1293,1757\}$$

possibly one further field; under the generalized Riemann hypothesis (GRH) — the Riemann hypothesis for all Dirichlet $L$-functions — at most one further such field exists, and if the exceptional field exists its discriminant is enormous (beyond $6.3\times 10^{16}$ by the quantitative bounds in this circle). The potential exception is not any identified small candidate: it is the single discriminant that Tatuzawa's ineffective estimate cannot exclude.

From 2003, Biró introduced a Riemann-hypothesis-free technique: it studies the special values at $s=0$ of partial Hecke $L$-functions attached to ideals of the field, which across a parametric family satisfy linear relations whose coefficients can be computed; an assumed class number one forces such values into an impossibly rigid pattern. With this he proved Chowla's conjecture ($d=4m^{2}+1$: only $m\in\{1,2,3,5,7,13\}$) and Yokoi's conjecture ($d=m^{2}+4$: only $m\in\{1,3,5,7,13,17\}$). Byeon–Kim–Lee settled Mollin's conjecture ($d=n^{2}-4$, $n$ odd squarefree case), Lee determined all ERD types with $d\not\equiv 5\pmod 8$, and Biró–Lapkova solved the two-parameter family $d=(an)^{2}+4a$ for odd $a,n$ (exactly the fourteen values $5,13,21,29,53,77,93,173,237,293,437,453,1133,1253$). A related elementary route, going back to Ankeny–Chowla–Hasse, shows $h(d)>1$ by proving that a Pell-type equation $x^{2}-dy^{2}=\pm 4p$ is unsolvable for a prime $p$ that must split into principal ideals when $h(d)=1$; Hoque and Kotyada used this to prove $h(d)>1$ for $d=a^{2}m^{2}+4ap$ ($a>1$, $p>2$ prime, $d$ squarefree). Each of these results covers part of the ERD family, but the residues $d\equiv 5\pmod 8$ outside the settled subfamilies (for instance $d=m^{2}+r$ with $m$ even and odd $r\mid m$ beyond Chowla's $r=1$, or negative $r=-4s$ with $s\mid m$, $s>1$) have resisted unconditional treatment, and the possible Tatuzawa exception has never been eliminated. Hoque–Kotyada (2021) note that finding — or ruling out — the exceptional ERD field without GRH is still an open problem, and the recent classification literature (Kopp–Lagarias 2025; Akbary–Maarefparvar 2025) continues to carry the 'with one possible exception' caveat unconditionally.

Problem Statement

Determine, without assuming the generalized Riemann hypothesis or any other unproved hypothesis, the complete list of the squarefree integers $d>1$ for which there exist integers $m\geq 1$ and $r$ with $d=m^{2}+r$ and $r\mid 4m$ — the extended Richaud–Degert (ERD) family, classically with $-m<r\leq m$ or $r=\pm 4m/3$ (allowing any divisor $r$ of $4m$ yields the same squarefree radicands $d>1$) — such that the real quadratic field $\mathbb{Q}(\sqrt{d})$ has class number one. Concretely: either prove unconditionally that the known 43 fields (the list in the background, due to Louboutin, Yokoi, and Mollin–Williams) exhaust the extended Richaud–Degert fields of class number one, or else exhibit the exceptional field — a specific ERD-type squarefree $d$, necessarily of enormous discriminant, with $h(d)=1$ — and prove that it is the only field beyond the 43. This is the problem left open by the 1990 analyses of Louboutin, Yokoi, and Mollin–Williams and restated by Hoque–Kotyada (2021) as 'finding this exceptional real quadratic field without generalized Riemann Hypothesis'; its resolution requires eliminating (or realizing) the single possible exception left by Tatuzawa's ineffective lower bound for $L(1,\chi)$ in the Mollin–Williams analysis.

References

1. Azizul Hoque and Srinivas Kotyada, 'Class number one problem for the real quadratic fields Q(√(m^2+2r))', Archiv der Mathematik 116(1), 33–36 (2021). DOI: 10.1007/s00013-020-01520-w; arXiv:2008.03505; https://doi.org/10.1007/s00013-020-01520-w
2. R. A. Mollin and H. C. Williams, 'Solution of the Class Number One Problem For Real Quadratic Fields of Extended Richaud-Degert Type (With One Possible Exception)', in Number Theory (Banff, AB, 1988), de Gruyter, Berlin, 417–426 (1990). DOI: 10.1515/9783110848632-033; https://doi.org/10.1515/9783110848632-033
3. Stéphane Louboutin, 'Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields', Canadian Journal of Mathematics 42(2), 315–341 (1990). DOI: 10.4153/CJM-1990-018-3; https://doi.org/10.4153/CJM-1990-018-3
4. Hideo Yokoi, 'The fundamental unit and class number one problem of real quadratic fields with prime discriminant', Nagoya Mathematical Journal 120, 51–59 (1990). DOI: 10.1017/S0027763000003238; https://doi.org/10.1017/S0027763000003238
5. András Biró, 'Yokoi's conjecture', Acta Arithmetica 106(1), 85–104 (2003). DOI: 10.4064/aa106-1-6; https://doi.org/10.4064/aa106-1-6
6. Dongho Byeon, Myoungil Kim and Jungyun Lee, 'Mollin's conjecture', Acta Arithmetica 126(2), 99–114 (2007). DOI: 10.4064/aa126-2-1; https://doi.org/10.4064/aa126-2-1
7. Jungyun Lee, 'The complete determination of wide Richaud–Degert types which are not 5 modulo 8 with class number one', Acta Arithmetica 140(1), 1–29 (2009). DOI: 10.4064/aa140-1-1; https://doi.org/10.4064/aa140-1-1
8. András Biró and Kostadinka Lapkova, 'The class number one problem for the real quadratic fields Q(√((an)^2+4a))', Acta Arithmetica 172(2), 117–131 (2016). DOI: 10.4064/aa7957-12-2015; https://doi.org/10.4064/aa7957-12-2015
9. Kalyan Chakraborty, Azizul Hoque and Mohit Mishra, 'A note on certain real quadratic fields with class number up to three', Kyushu Journal of Mathematics 74(1), 201–210 (2020). DOI: 10.2206/kyushujm.74.201; arXiv:1812.02488; https://doi.org/10.2206/kyushujm.74.201
10. N. C. Ankeny, S. Chowla and H. Hasse, 'On the class-number of the maximal real subfield of a cyclotomic field', Journal für die reine und angewandte Mathematik 217, 217–220 (1965). DOI: 10.1515/crll.1965.217.217; https://doi.org/10.1515/crll.1965.217.217
11. Gene S. Kopp and Jeffrey C. Lagarias, 'Unit-generated orders of real quadratic fields I. Class number bounds', preprint (2025). arXiv:2512.11311; https://arxiv.org/abs/2512.11311
12. Amir Akbary and Abbas Maarefparvar, 'Number Fields With Large Pólya Groups', preprint (2025). arXiv:2508.11125; https://arxiv.org/abs/2508.11125