Non-trivial plus parts of cyclotomic class numbers for continued-fraction families with $d \equiv 2 \pmod 4$

For a positive integer $n$, let $h_n^+$ denote the class number of the maximal real subfield $\mathbb{Q}(\zeta_n)^+ = \mathbb{Q}(\zeta_n + \zeta_n^{-1})$ of the cyclotomic field $\mathbb{Q}(\zeta_n)$ (the plus part of the cyclotomic class number), an invariant for which no general effective method of computation is known. Prakash (arXiv 2022; Indian J. Pure Appl. Math. 57, 2026) claimed that $h_{4d}^+ > 1$ for every square-free member of the general Dahl–Kala continued-fraction families of real quadratic discriminants with $d \equiv 3 \pmod 4$, and explicitly excluded the residue class $d \equiv 2 \pmod 4$, leaving it open; the lifting step also excludes the finitely many $d$ with $\varphi(d) \le 4$. Chakraborty and Hoque (Chinese Ann. Math. Ser. B 46, 2025) subsequently proved $h_{4m}^+ > 1$ for three isolated Richaud–Degert-type families with $m \equiv 2 \pmod 4$ (namely $m = (14(2n+1))^2 + 2$, $m = (3(2n+1))^2 + 1$, $m = (6(2n+1))^2 - 2$). The surviving open core is the general continued-fraction-family version: prove, or disprove, that $h_{4d}^+ > 1$ for the square-free members $d \equiv 2 \pmod 4$ of the admissible Dahl–Kala families, with the finitely many small-conductor members ($d \in \{2, 6, 10\}$) excluded from the claim, going beyond the known isolated families.

Background

For a number field $K$ (a finite extension of the rationals $\mathbb{Q}$), the ideal class group $\mathrm{Cl}(K)$ of the ring of integers $\mathcal{O}_K$ is a finite abelian group measuring the failure of unique factorization of integers in $\mathcal{O}_K$; its order $h(K)$ is the class number, and $h(K) = 1$ exactly when $\mathcal{O}_K$ has unique factorization into irreducible elements.

For $n \ge 1$ let $\zeta_n$ be a primitive $n$-th root of unity. The cyclotomic field $\mathbb{Q}(\zeta_n)$ has degree $\varphi(n)$ over $\mathbb{Q}$, where $\varphi$ is Euler's totient function, and its maximal real subfield is the fixed field of complex conjugation, $\mathbb{Q}(\zeta_n)^+ = \mathbb{Q}(\zeta_n + \zeta_n^{-1})$, of degree $\varphi(n)/2$ for $n \ge 3$. The class number $h_n$ of $\mathbb{Q}(\zeta_n)$ factors as $h_n = h_n^- \cdot h_n^+$, where the plus part $h_n^+$ is the class number of $\mathbb{Q}(\zeta_n)^+$ and the minus part (relative class number) is $h_n^- = h_n / h_n^+$. The minus part is effectively computable: the analytic class number formula expresses it as a finite product over odd Dirichlet characters $\chi$ (the multiplicative characters attached to the field with $\chi(-1) = -1$) of generalized Bernoulli numbers $B_{1,\chi}$. The plus part is, by contrast, one of the most enigmatic invariants in algebraic number theory: Minkowski-type discriminant bounds and Odlyzko's discriminant bounds become ineffective at large conductors, and $h_n^+$ is known exactly only for small conductors — unconditionally for prime conductors up to 151 and up to 241 under the generalized Riemann hypothesis (GRH), as summarized by Chakraborty and Hoque (2025).

One route to exhibiting $h_n^+ > 1$ is through explicit quadratic families. Ankeny, Chowla and Hasse (1965) proved $h_p^+ > 1$ when $p = (2nq)^2 + 1$ is prime with $q$ prime and $n > 1$; S. D. Lang (1977) proved the same for primes $p = ((2n+1)q)^2 + 4$; Osada (1987) extended both results to suitable square-free integers of these shapes. These arguments rest on a lifting principle: Yamaguchi (1975) proved that for a positive integer $d$ with $\varphi(d) > 4$, the class number of the real quadratic field $\mathbb{Q}(\sqrt d)$ divides $h_{4d}^+$ (this is the form in which both Hoque–Chakraborty (2018) and Prakash quote and use it); Osada proved a refinement (Nagoya Math. J. 113, 1989), quoted by Chakraborty and Hoque (2025) as their Lemma 2.2: for square-free $m$, the class group of $\mathbb{Q}(\zeta_{\sigma_m^2 m})^+$ contains a subgroup isomorphic to the $\sigma_m^2$-th powers of the class group of $\mathbb{Q}(\sqrt m)$, where $\sigma_m = 1$ if $m \equiv 1 \pmod 4$ and $\sigma_m = 2$ if $m \equiv 2, 3 \pmod 4$. In particular, by Yamaguchi's divisibility, proving $h(\mathbb{Q}(\sqrt d)) > 1$ forces $h_{4d}^+ > 1$ whenever $\varphi(d) > 4$, for $d \equiv 2 \pmod 4$ as well as for $d \equiv 3 \pmod 4$.

The essential step is thus proving non-triviality of the quadratic class number, and the standard technique is Pell-type: if a rational prime $p$ splits in $\mathbb{Q}(\sqrt d)$, i.e. the Legendre symbol $(d/p) = 1$ ($d$ is a nonzero square modulo $p$), then $p\mathcal{O} = \mathcal{P} \mathcal{P}'$ with prime ideals of norm $p$; if moreover the norm equation $x^2 - dy^2 = \pm p$ has no integer solution, then $\mathcal{P}$ is not principal and $h(\mathbb{Q}(\sqrt d)) > 1$. Hoque and Chakraborty (2018) applied this with a prime built into the parametrization: for $d = (2np)^2 - 1$ with $p \equiv 1 \pmod 4$ prime one has $d \equiv -1 \pmod p$, so $p$ splits by quadratic reciprocity, and a descent using the fundamental unit rules out the equation; they obtained four explicit families with $d \equiv 3 \pmod 4$ and $h_{4d}^+ > 1$ (with related families of the same type due to Hoque and Saikia, 2016).

The stage for the present problem is the systematic supply of such families from continued fractions. For square-free $d$, the continued fraction of $\sqrt d$ is periodic with a symmetric (palindromic) period. The converse construction recorded by Dahl and Kala (2018, their Proposition 6, following Halter-Koch) states: given a symmetric sequence $u_1, \dots, u_{s-1}$ of positive integers ($s \ge 1$; the sequence is empty for $s = 1$), define continuants by $q_{i+1} = u_{i+1} q_i + q_{i-1}$ from $q_{-1} = 0$, $q_0 = 1$; then the equation $\sqrt d = [k, \overline{u_1, \dots, u_{s-1}, 2k}]$ with $k = \lfloor \sqrt d \rfloor$ has infinitely many square-free solutions $d \equiv 2, 3 \pmod 4$ if and only if $q_{s-1}$ is odd or $q_{s-1}$ and the product $q_{s-2} q_{s-3}$ are both even (equivalently, since consecutive continuants cannot both be even, if and only if $q_{s-1}$ is odd or $q_{s-1}$ and $q_{s-3}$ are both even). All solutions are of the form $d = d(n) = \alpha n^2 + \beta n + \gamma$ with $\alpha \ne 0$ a perfect square and discriminant $\beta^2 - 4\alpha\gamma \in \{(-1)^s, 4(-1)^s, 16(-1)^s\}$, and for the square-free solutions with $d \equiv 2, 3 \pmod 4$ the fundamental unit of $\mathbb{Q}(\sqrt d)$ (the smallest unit $> 1$ generating the unit group up to sign) is $\varepsilon = p(n) + q\sqrt d$ with $p(n)$ linear in $n$, $q = q_{s-1}$ fixed, and norm $p(n)^2 - q^2 d(n) = (-1)^s$. A parity computation with this norm identity modulo $4$ shows that the square-free members of a family with $q_{s-1}$ odd all lie in a single residue class, namely $d \equiv 3 \pmod 4$ when the period length $s$ is even and $d \equiv 2 \pmod 4$ when $s$ is odd, while families with $q_{s-1}$, $q_{s-3}$ both even can have members in both classes, the class depending only on $n \bmod 4$. The simplest example, the sequence $u = (1)$, gives $d = k^2 + 2k = (k+1)^2 - 1$ and contains the Hoque–Chakraborty fields $(2np)^2 - 1$.

Prakash (arXiv 2022; published Indian J. Pure Appl. Math. 57, 2026) applied the Pell-type technique to these general families and announced that for the sub-case producing $d \equiv 3 \pmod 4$ (even period length, $q_{s-1}$ odd, fundamental unit of norm $+1$) one has $h_{4d}^+ > 1$ for every square-free family member, subsuming the 2018 families. His treatment explicitly leaves aside the residue class $d \equiv 2 \pmod 4$ ("we aren't interested in $d \equiv 2 \pmod 4$"), and the lifting lemma excludes the finitely many $d$ with $\varphi(d) \le 4$. In a complementary direction, Mishra, Schoof and Washington (2021) proved existentially, by class field theory without explicit families, that every finite abelian group occurs as a subgroup of the class group of infinitely many real cyclotomic fields. Most relevantly, Chakraborty and Hoque (2025) then took up the class $m \equiv 2 \pmod 4$, stating that it "would be interesting to try exhibiting similar families" in it: they proved $h_{4m}^+ > 1$ for the square-free members of three isolated Richaud–Degert-type families — $m = (14(2n+1))^2 + 2$, $m = (3(2n+1))^2 + 1$ and $m = (6(2n+1))^2 - 2$ (Richaud–Degert fields are the real quadratic fields $\mathbb{Q}(\sqrt{N^2 + r})$ with a fixed small nonzero integer $r$, studied through the explicit continued fraction of $\sqrt{N^2 + r}$) — via the Pell technique together with Osada's lifting, and also produced a family whose plus part is divisible by 3. What no work in this lineage does is treat the general continued-fraction families in the $d \equiv 2 \pmod 4$ residue class.

Problem Statement

Let $s \ge 1$ and let $u_1, \dots, u_{s-1}$ be a symmetric sequence of positive integers (empty when $s = 1$), with continuants defined by $q_{i+1} = u_{i+1} q_i + q_{i-1}$ ($q_{-1} = 0$, $q_0 = 1$). Call the sequence admissible if $q_{s-1}$ is odd, or $q_{s-1}$ and the product $q_{s-2} q_{s-3}$ are both even (equivalently, since consecutive continuants cannot both be even: $q_{s-1}$ is odd, or $q_{s-1}$ and $q_{s-3}$ are both even). For an admissible sequence, the equation

$\sqrt{d} = [k, \overline{u_1, \dots, u_{s-1}, 2k}], \qquad k = \lfloor \sqrt d \rfloor,$

has, by the Dahl–Kala converse construction, infinitely many square-free solutions $d \equiv 2, 3 \pmod 4$, all of the form $d = d(n) = \alpha n^2 + \beta n + \gamma$ with $\alpha \ne 0$ a perfect square and discriminant $\beta^2 - 4\alpha\gamma \in \{(-1)^s, 4(-1)^s, 16(-1)^s\}$; for the square-free solutions with $d \equiv 2, 3 \pmod 4$ the fundamental unit of $\mathbb{Q}(\sqrt d)$ is $\varepsilon = p(n) + q\sqrt d$ with $p(n)$ linear in $n$, $q = q_{s-1}$, and norm $p(n)^2 - q^2 d(n) = (-1)^s$. By this norm identity modulo $4$, the square-free members of a family with $q_{s-1}$ odd all lie in a single residue class — $d \equiv 3 \pmod 4$ if $s$ is even (Prakash's case, norm of the fundamental unit $+1$) and $d \equiv 2 \pmod 4$ if $s$ is odd (norm $-1$) — while admissible sequences with $q_{s-1}$, $q_{s-3}$ both even can contribute members in both classes, the class depending only on $n \bmod 4$.

Prove, or disprove: for every admissible sequence and every $n$ for which $d(n)$ is square-free and $d(n) \equiv 2 \pmod 4$ — excluding the members with $\varphi(d) \le 4$, i.e. $d \in \{2, 6, 10\}$ in this residue class, of conductors $8, 24, 40$ — the class number of the maximal real subfield $\mathbb{Q}(\zeta_{4d(n)})^+$ of the cyclotomic field $\mathbb{Q}(\zeta_{4d(n)})$ satisfies $h^+(\mathbb{Q}(\zeta_{4d(n)})) > 1$, where $h^+$ denotes the class number of the maximal real subfield.

An affirmative resolution establishes the non-triviality for the general admissible families in the residue class $d \equiv 2 \pmod 4$ — the exact residue-class analogue of Prakash's theorem for $d \equiv 3 \pmod 4$, and a statement which, applied to the admissible family containing it, covers the isolated family $m = (3(2n+1))^2 + 1$ already settled by Chakraborty and Hoque (2025) (a subfamily of the admissible family with empty interior sequence, $s = 1$, $\sqrt{m} = [k, \overline{2k}]$ with $k = 3(2n+1)$); their other two families, $m = (14(2n+1))^2 + 2$ and $m = (6(2n+1))^2 - 2$, lie outside the scope of this statement, since with $N = 14(2n+1)$ and $N = 6(2n+1)$ respectively their members have $\sqrt{m} = [N, \overline{N, 2N}]$ and $\sqrt{m} = [N-1, \overline{1, N-2, 1, 2N-2}]$ (e.g. $\sqrt{34} = [5, \overline{1, 4, 1, 10}]$), whose interior partial quotients ($u_1 = N$ and $u = (1, N-2, 1)$) depend on the parameter $N$ and so are not the fixed sequence $u_1, \dots, u_{s-1}$ of any admissible family. A refutation exhibits an admissible sequence together with a member $d \equiv 2 \pmod 4$ for which the claim fails.

References

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