Family: thue-mahler — Thue-Mahler equation

diophantine_classifier/data/families/thue-mahler.yaml

@@ -0,0 +1,26 @@
+# Family: Thue-Mahler 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: thue-mahler
+name: Thue-Mahler equation
+priority: 1
+status: algorithmic
+class: curve
+form: F(x, y) = m * p_1^N_1 * ... * p_s^N_s
+parents:
+- thue
+matcher: true
+methods:
+- S-unit reduction (Tzanakis-de Weger)
+- Gherga-Siksek algorithm
+software:
+  magma: Gherga-Siksek ThueMahler (GitHub)
+examples:
+- x^3 + 2*y^3 = 5^a * 11^b
+references:
+- key: Mahler1933
+  why: "origin of the problem: finiteness with prime-power right-hand sides (p-adic method)"
+- key: TzanakisDeWeger1992
+  why: the classical practical algorithm
+- key: GhergaSiksek2022
+  why: the modern efficient implementation (Magma code)

docs/FAMILIES.md

@@ -177,6 +177,15 @@ certification for large fields); Magma: `Thue`.
 **References.** Thue 1909; Baker 1968; Bilu–Hanrot, "Solving Thue equations of high
 degree" (J. Number Theory 60, 1996).
 
+### `thue-mahler` — Thue–Mahler equation — P1, effective/algorithmic
+**Form.** F(x, y) = m · p₁^{N₁} ⋯ p_s^{N_s}, F irreducible deg ≥ 3, gcd(x,y) restrictions.
+**Status.** Finite (Mahler 1933), effective (Coates 1969); practical algorithm
+Tzanakis–de Weger 1992; modern efficient implementation Gherga–Siksek.
+**Software.** Magma: Gherga–Siksek `ThueMahler` code (GitHub); PARI ≥ 2.17 has
+S-unit tooling to script it.
+**References.** Mahler 1933; Tzanakis–de Weger 1992; Gherga–Siksek,
+"Efficient resolution of Thue–Mahler equations" (2022).
+
 ### `hyperelliptic` — Hyperelliptic curves — P1, effective (integral) / ineffective (rational)
 **Form.** y² = f(x), f squarefree, deg f ≥ 5.
 **Status.** Integral points: finite, effective (Baker); practical via Baker + LLL

tests/test_classify.py

@@ -40,6 +40,7 @@
     ("x^4 + y^4 + z^4 = w^4", "", "equal-sums-like-powers"),   # Elkies
     # polynomial-exponential
     ("x^3 - 4 = y^n", "", "power-values"),
+    ("x^3 + 2*y^3 = 5^a * 11^b", "", "thue-mahler"),
     # unit fractions
     ("1/x + 1/y + 1/z = 1", "", "egyptian-fractions"),
 ]