Conjectures associated with A038552

The question

mod_24_of_isA038552. All terms of A038552 are congruent to $19(\mathrm{mod}24)$.

isA038552_eq_largestNegFundDisc. A038552 also gives the largest absolute value of negative fundamental discriminant for each class number.

largestEven_lt_largestOdd_negFundDisc. For even class number $n$, let $k$ be the largest odd number such that the quadratic field with discriminant $-k$ has class number $n$, and let $k^{\prime }$ be the largest even such number, when they exist. Then $k>k^{\prime }$.

A038552 lists the largest squarefree number $k$ such that the imaginary quadratic field $\mathbb{Q}(\sqrt{-k})$ has class number $n$.

The conjectures state that:

1. All terms are congruent to $19(\mathrm{mod}24)$.
2. This is also the largest absolute value of negative fundamental discriminant $d$ for class number $n$.
3. For even $n$, if $k$ is the largest odd number with $h(-k)=n$ and $k^{\prime }$ is the largest even number with $h(-k^{\prime })=n$, then $k>k^{\prime }$. Here $h(D)$ is the class number of the quadratic field with discriminant $D$, so $k$ and $k^{\prime }$ are absolute values of negative fundamental discriminants, not radicands. The source states conjecture 2 in this form.

The squarefree condition in the definition is needed for the maximum to exist, since $\mathbb{Q}(\sqrt{-k})=\mathbb{Q}(\sqrt{-4k})$.

Conjecture 2 is not a restatement of the definition. Both maxima range over the same imaginary quadratic fields, but they maximize different integers attached to those fields. A038552 uses the squarefree radicand $k$, whereas the discriminant of $\mathbb{Q}(\sqrt{-k})$ is $-k$ for $k\equiv 3(\mathrm{mod}4)$ and $-4k$ otherwise. The map $k\mapsto |d|$ is not monotone: it sends $2$ to $8$ and $3$ to $3$. So conjecture 2 says that the largest term $k$ satisfies $k\equiv 3(\mathrm{mod}4)$, and that $4k^{\prime }\le k$ for every $k^{\prime }\equiv 1,2(\mathrm{mod}4)$ with class number $n$.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«38552» (3 statements).

theorem mod_24_of_isA038552 {n k : ℕ} (h : IsA038552 n k) : k % 24 = 19

theorem isA038552_eq_largestNegFundDisc {n k : ℕ} (h : IsA038552 n k) :
    IsLargestNegFundDiscrForClassNumber (n := n) k

theorem largestEven_lt_largestOdd_negFundDisc {n k k' : ℕ} (hn : Even n)
    (hk : IsGreatest {m : ℕ | Odd m ∧ IsFundamentalDiscr (-m : ℤ) ∧
      classNumberOfDiscriminant (-m : ℤ) = n} k)
    (hk' : IsGreatest {m : ℕ | Even m ∧ IsFundamentalDiscr (-m : ℤ) ∧
      classNumberOfDiscriminant (-m : ℤ) = n} k') : k' < k

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• [Sta67] Stark, Harold M. "A complete determination of the complex quadratic fields of class-number one." Michigan Mathematical Journal 14.1 (1967): 1-27.
• oeis.org/A038552

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.