Family: catalan — Catalan equation

diophantine_classifier/data/families/catalan.yaml

@@ -0,0 +1,24 @@
+# Family: Catalan 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: catalan
+name: Catalan equation
+priority: 1
+status: solved
+class: expdioph
+form: x^p - y^q = 1, x, y, p, q >= 2
+parents:
+- exponential-diophantine
+matcher: true
+methods:
+- cyclotomic fields (Mihailescu)
+notes: Only solution 3^2 - 2^3 = 1.
+examples:
+- x^p - y^q = 1
+references:
+- key: Mihailescu2004
+  why: the complete solution of Catalan's conjecture
+- key: Tijdeman1976
+  why: the earlier effective finiteness via Baker's method
+- key: BiluBugeaudMignotte2014
+  why: book exposition of the proof

diophantine_classifier/solvers.py

@@ -1154,6 +1154,25 @@ def rec(k_left, target, minimum, acc):
         "permutations give the rest", complete=True)
 
 
+def _solve_catalan(cls, match):
+    r"""
+    Catalan's equation: Mihailescu's theorem.
+
+    EXAMPLES::
+
+        sage: from diophantine_classifier import solve
+        sage: solve("x^p - y^q = 1").solutions    # unknowns (x, p, y, q)
+        [(3, 2, 2, 3)]
+    """
+    # 3^2 - 2^3 = 1 in the standard coordinates (x, p, y, q)
+    standard = {"x": ZZ(3), "p": ZZ(2), "y": ZZ(2), "q": ZZ(3)}
+    names = _normalized(match)
+    return SolutionSet(
+        names, [tuple(standard[v] for v in names)], "finite-complete",
+        "Mihailescu's theorem: 3^2 - 2^3 = 1 is the only solution in "
+        "integers > 1", complete=True)
+
+
 SOLVERS = {
     "univariate": _solve_univariate,
     "linear": _solve_linear,
@@ -1164,6 +1183,7 @@ def rec(k_left, target, minimum, acc):
     "elliptic-weierstrass": _solve_weierstrass,
     "thue": _solve_thue,
     "egyptian-fractions": _solve_egyptian,
+    "catalan": _solve_catalan,
 }
 
 

docs/FAMILIES.md

@@ -243,6 +243,14 @@ primitive solutions with min ≥ 2 exponents; Fermat–Catalan/Beal conjectures
 Poonen–Schaefer–Stoll 2007; Bennett–Chen–Dahmen–Yazdani, "Generalized Fermat
 equations: a miscellany" (Int. J. Number Theory 11, 2015).
 
+### `catalan` — Catalan's equation — P1, solved
+**Form.** x^p − y^q = 1, x, y > 0, p, q ≥ 2.
+**Status.** Only 3² − 2³ = 1 (Mihăilescu 2002/2004, published Crelle 2004), using
+cyclotomic fields — no logarithm bounds needed. Tijdeman 1976 had given effective
+finiteness.
+**References.** Catalan 1844; Tijdeman 1976; Mihăilescu 2004; Bilu–Bugeaud–Mignotte,
+*The Problem of Catalan* (2014).
+
 
 ---
 

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^p - y^q = 1", "", "catalan"),
     # unit fractions
     ("1/x + 1/y + 1/z = 1", "", "egyptian-fractions"),
 ]

tests/test_solvers.py

@@ -74,6 +74,13 @@ def test_thue():
     assert_valid_solutions("x^3 + 2*y^3 = 11", s.solutions)
 
 
+def test_catalan():
+    s = solve("x^p - y^q = 1")
+    # unknowns ordered (x, p, y, q)
+    assert s.solutions == [(3, 2, 2, 3)]
+    assert s.complete
+
+
 def test_unavailable_carries_hints():
     with pytest.raises(SolverUnavailable) as err:
         solve("y^2 = x^7 + 3")