From 831c0f9675f2e1db0ee498c1c396bac3002d081d Mon Sep 17 00:00:00 2001
From: David Roe <roed.math@gmail.com>
Date: Sun, 9 Aug 2026 16:26:21 -0400
Subject: [PATCH] =?UTF-8?q?Family:=20nagell-ljunggren=20=E2=80=94=20Nagell?=
 =?UTF-8?q?-Ljunggren=20equation?=
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One family: **Nagell-Ljunggren equation** (`nagell-ljunggren`) — P2, open.

- Standard form: `(x^n - 1)/(x - 1) = y^q`
- Parent: `exponential-diophantine`

Contents: the registry entry and its `docs/FAMILIES.md` section.

References cited (the claim each one supports):

- `Ljunggren1943` — the classical partial results
- `BugeaudMignotte2002` — survey of the state of the art

Independent of every other family PR: it needs only the registry backbone it is based on.

Part of the series that splits #1 into reviewable pieces:

1. `01-bibliography` — packaging, docs, annotated bibliography
2. `02-parsing` — equation strings to a term model
3. `03-registry` — the YAML family registry (3 seed families)
4. `04-matchers` — shape recognizers
5. `05-classify` — the classification pipeline
6. `06-solvers` — solver framework, two seed solvers, and the CLI
7. `07..09-backbone` — the 23 parent families of the DAG, by depth
8. one PR per remaining family (38 of them, mutually independent)
9. `99-polish` — restore the full doctests and tighten the invariants
---
 .../data/families/nagell-ljunggren.yaml        | 18 ++++++++++++++++++
 docs/FAMILIES.md                               |  8 ++++++++
 2 files changed, 26 insertions(+)
 create mode 100644 diophantine_classifier/data/families/nagell-ljunggren.yaml

diff --git a/diophantine_classifier/data/families/nagell-ljunggren.yaml b/diophantine_classifier/data/families/nagell-ljunggren.yaml
new file mode 100644
index 0000000..7a78a03
--- /dev/null
+++ b/diophantine_classifier/data/families/nagell-ljunggren.yaml
@@ -0,0 +1,18 @@
+# Family: Nagell-Ljunggren equation
+# Schema: see data/families/README.md; prose documentation in docs/FAMILIES.md.
+# References are keys into data/references.bib, each with a 'why' annotation.
+slug: nagell-ljunggren
+name: Nagell-Ljunggren equation
+priority: 2
+status: open
+class: expdioph
+form: (x^n - 1)/(x - 1) = y^q
+parents:
+- exponential-diophantine
+matcher: false
+notes: 'Known: (3,5,11,2), (7,4,20,2), (18,3,7,3); conjecturally all.'
+references:
+- key: Ljunggren1943
+  why: the classical partial results
+- key: BugeaudMignotte2002
+  why: survey of the state of the art
diff --git a/docs/FAMILIES.md b/docs/FAMILIES.md
index 86434cf..a1e8829 100644
--- a/docs/FAMILIES.md
+++ b/docs/FAMILIES.md
@@ -257,6 +257,14 @@ perfect power).
 **References.** Schinzel–Tijdeman 1976; Erdős–Selfridge 1975; Shorey–Tijdeman,
 *Exponential Diophantine Equations* (1986) — the standard reference for this whole section.
 
+### `nagell-ljunggren` — Nagell–Ljunggren equation — P2, open
+**Form.** (xⁿ − 1)/(x − 1) = y^q, x, y ≥ 2, n ≥ 3, q ≥ 2.
+**Status.** Three known solutions (x,n,y,q) = (3,5,11,2), (7,4,20,2), (18,3,7,3);
+conjecturally all; many partial results (Ljunggren: q = 2 solved; Bugeaud–Mignotte
+surveys); finiteness unknown in general.
+**References.** Nagell 1920; Ljunggren 1943; Bugeaud–Mignotte,
+"L'équation de Nagell–Ljunggren" (Enseign. Math. 48, 2002).
+
 ### `exponential-diophantine` — Purely exponential equations — P2, effective (few terms)
 **Form.** c₁·b₁^{n₁} + ⋯ + c_k·b_k^{n_k} = c (fixed bases, unknown exponents);
 e.g. 2ᵃ + 3ᵇ = 5ᶜ, Goormaghtigh-type, Jeśmanowicz conjecture instances.