Family: nagell-ljunggren — Nagell-Ljunggren equation

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

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.