feat(OEIS): A038552

FormalConjectures/OEIS/38552.lean

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+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+import FormalConjectures.Util.ProblemImports
+
+open Polynomial
+
+/-!
+# Conjectures associated with A038552
+
+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 $19 \pmod{24}$.
+2. This is also the largest absolute value of negative fundamental discriminant $d$ for
+   class number $n$. For even $n$, if $k$ is the largest odd number with $h(-k) = n$ and
+   $k'$ is the largest even number with $h(-k') = n$, then $k > k'$.
+
+*References:* [oeis.org/A038552](https://oeis.org/A038552)
+-/
+
+namespace OeisA38552
+
+/-- The class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-k})$ equals $n$. -/
+def HasClassNumber (k n : ℕ) : Prop :=
+  ∃ (h : Irreducible (X ^ 2 + C (k : ℚ))),
+  haveI := Fact.mk h
+  NumberField.classNumber (AdjoinRoot (X ^ 2 + C (k : ℚ))) = n
+
+/-- $k$ is the largest squarefree number such that $\mathbb{Q}(\sqrt{-k})$ has class number $n$.
+This defines the $n$-th term of A038552. -/
+def a (n k : ℕ) : Prop :=
+  Squarefree k ∧ HasClassNumber k n ∧
+  ∀ m, Squarefree m → HasClassNumber m n → m ≤ k
+
+/-- An integer $d < 0$ is a negative fundamental discriminant if either:
+- $d \equiv 1 \pmod 4$ and $d$ is squarefree, or
+- $d = 4m$ where $m \equiv 2$ or $3 \pmod 4$ and $m$ is squarefree. -/
+def IsNegFundamentalDiscriminant (d : ℤ) : Prop :=
+  d < 0 ∧ ((d % 4 = 1 ∧ Squarefree d) ∨
+           (∃ m : ℤ, d = 4 * m ∧ (m % 4 = 2 ∨ m % 4 = 3) ∧ Squarefree m))
+
+/-- The class number of the quadratic field with discriminant $d$. -/
+noncomputable def classNumberOfDiscriminant (d : ℤ) : ℕ :=
+  haveI := Classical.dec (Irreducible (X ^ 2 - C (d : ℚ)))
+  if h : Irreducible (X ^ 2 - C (d : ℚ)) then
+    haveI := Fact.mk h
+    NumberField.classNumber (AdjoinRoot (X ^ 2 - C (d : ℚ)))
+  else 0
+
+/-- $|d|$ is the largest absolute value among negative fundamental discriminants
+with class number $n$. -/
+def IsLargestNegFundDiscForClassNumber {n : ℕ} (absD : ℕ) : Prop :=
+  IsNegFundamentalDiscriminant (-(absD : ℤ)) ∧
+  classNumberOfDiscriminant (-(absD : ℤ)) = n ∧
+  ∀ d : ℤ, IsNegFundamentalDiscriminant d → classNumberOfDiscriminant d = n → d.natAbs ≤ absD
+
+/-- The Stark-Heegner theorem implies that the squarefree $k > 0$ such that
+$\mathbb{Q}(\sqrt{-k})$ has class number $1$ are exactly $\{1, 2, 3, 7, 11, 19, 43, 67, 163\}$. -/
+@[category research solved, AMS 11]
+theorem starkHeegner_classNumberOne :
+    {k : ℕ | Squarefree k ∧ HasClassNumber k 1} = {1, 2, 3, 7, 11, 19, 43, 67, 163} := by
+  sorry
+
+/-- $163$ is squarefree. -/
+@[category test, AMS 11]
+theorem squarefree_163 : Squarefree (163 : ℕ) :=
+  (Nat.prime_iff.mp (by norm_num : Nat.Prime 163)).squarefree
+
+/-- $\mathbb{Q}(\sqrt{-163})$ has class number $1$. -/
+@[category API, AMS 11]
+theorem hasClassNumber_163_1 : HasClassNumber 163 1 := by
+  have h := starkHeegner_classNumberOne
+  simp only [Set.ext_iff, Set.mem_setOf_eq, Set.mem_insert_iff, Set.mem_singleton_iff] at h
+  exact ((h 163).mpr (by right; right; right; right; right; right; right; right; rfl)).2
+
+/-- $163$ is the largest squarefree $k$ with class number $1$. -/
+@[category test, AMS 11]
+theorem a_1_163 : a 1 163 := by
+  refine ⟨squarefree_163, hasClassNumber_163_1, ?_⟩
+  intro m hm_sq hm_class
+  have h := starkHeegner_classNumberOne
+  simp only [Set.ext_iff, Set.mem_setOf_eq, Set.mem_insert_iff, Set.mem_singleton_iff] at h
+  have hm_in : m ∈ ({1, 2, 3, 7, 11, 19, 43, 67, 163} : Set ℕ) := (h m).mp ⟨hm_sq, hm_class⟩
+  simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hm_in
+  rcases hm_in with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl <;> norm_num
+
+/-!
+The values for other class numbers in A038552 come from the papers
+* Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields,
+Math. Comp., 31 (1977), 786-796.
+* M. Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation 73 (2004),
+pp. 907-938.
+-/
+
+/-- All terms of A038552 are congruent to $19 \pmod{24}$. -/
+@[category research open, AMS 11]
+theorem mod_24_of_a {n k : ℕ} (h : a n k) : k % 24 = 19 := by
+  sorry
+
+/-- A038552 also gives the largest absolute value of negative fundamental discriminant
+for each class number. -/
+@[category research open, AMS 11]
+theorem a_eq_largestNegFundDisc {n k : ℕ} (h : a n k) :
+  IsLargestNegFundDiscForClassNumber (n := n) k := by
+  sorry
+
+/-- For even class number $n$, if $k$ is the largest odd squarefree number with
+$h(-k) = n$ and $k'$ is the largest even squarefree number with $h(-k') = n$,
+then $k > k'$. -/
+@[category research open, AMS 11]
+theorem odd_gt_even_for_even_classNumber {n : ℕ} (hn : Even n) (k k' : ℕ)
+    (hk_odd : Odd k) (hk'_even : Even k')
+    (hk_sq : Squarefree k) (hk'_sq : Squarefree k')
+    (hk_class : HasClassNumber k n) (hk'_class : HasClassNumber k' n)
+    (hk_largest : ∀ m, Odd m → Squarefree m → HasClassNumber m n → m ≤ k)
+    (hk'_largest : ∀ m, Even m → Squarefree m → HasClassNumber m n → m ≤ k') :
+    k > k' := by
+  sorry
+
+end OeisA38552