Family: s-unit — S-unit equation

diophantine_classifier/data/families/s-unit.yaml

@@ -0,0 +1,21 @@
+# Family: S-unit 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: s-unit
+name: S-unit equation
+priority: 2
+status: algorithmic
+class: expdioph
+form: a*u + b*v = c, u, v S-units
+parents:
+- exponential-diophantine
+matcher: true
+methods:
+- Baker + LLL (de Weger)
+software:
+  sage: K.solve_S_unit_equation(S) (number fields)
+references:
+- key: deWeger1987
+  why: the practical LLL-based algorithm
+- key: Alvarado2021
+  why: the Sage implementation for arbitrary number fields

docs/FAMILIES.md

@@ -257,6 +257,16 @@ perfect power).
 **References.** Schinzel–Tijdeman 1976; Erdős–Selfridge 1975; Shorey–Tijdeman,
 *Exponential Diophantine Equations* (1986) — the standard reference for this whole section.
 
+### `s-unit` — S-unit equations — P2, algorithmic
+**Form.** ax + by = c with x, y S-units (over ℤ: ± products of fixed primes;
+generally in a number field K with finite S).
+**Status.** Finite (Siegel–Mahler), effective (Baker); practical algorithm
+de Weger 1987 (LLL); implemented in Sage for arbitrary K, S. The workhorse
+that many other families reduce to (Thue–Mahler, curves via étale covers).
+**Software.** Sage: `K.solve_S_unit_equation(S)`; Magma: S-unit machinery.
+**References.** Mahler 1933; de Weger 1987; Evertse–Győry, *Unit Equations* (2015);
+Alvarado et al. 2019 (Sage implementation).
+
 ### `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.

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"),
+    ("2^a + 3^b = 5^c", "", "s-unit"),
     # unit fractions
     ("1/x + 1/y + 1/z = 1", "", "egyptian-fractions"),
 ]