Family: mordell — Mordell equation

diophantine_classifier/data/families/mordell.yaml

@@ -0,0 +1,27 @@
+# Family: Mordell 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: mordell
+name: Mordell equation
+priority: 1
+status: algorithmic
+class: genus1
+form: y^2 = x^3 + k
+parents:
+- elliptic-weierstrass
+matcher: true
+lmfdb: ec.q
+methods:
+- elliptic logarithms
+- tabulated for |k| <= 10^7
+software:
+  sage: EllipticCurve([0,0,0,0,k]).integral_points()
+code:
+  sage: EllipticCurve([0, 0, 0, 0, {k}]).integral_points(both_signs=True)
+examples:
+- y^2 = x^3 - 2
+references:
+- key: GebelPethoZimmer1998
+  why: the method, with tables for |k| <= 10^4
+- key: BennettGhadermarzi2015
+  why: complete tables for |k| <= 10^7

diophantine_classifier/solvers.py

@@ -1161,6 +1161,7 @@ def rec(k_left, target, minimum, acc):
     "binary-qf-representation": _solve_bqf,
     "quadratic-form-zero": _solve_qf_zero,
     "legendre": _solve_legendre,
+    "mordell": _solve_weierstrass,
     "elliptic-weierstrass": _solve_weierstrass,
     "thue": _solve_thue,
     "egyptian-fractions": _solve_egyptian,

docs/FAMILIES.md

@@ -146,6 +146,16 @@ by the elliptic-logarithm method once generators are known.
 **References.** Mordell 1922; Siegel 1929; Baker 1968; Gebel–Pethő–Zimmer 1994;
 Stroeker–Tzanakis 1994; Cremona, *Algorithms for Modular Elliptic Curves*.
 
+### `mordell` — Mordell equation — P1, algorithmic*
+**Form.** y² = x³ + k, k ≠ 0.
+**Status.** Finitely many integral points (Mordell/Siegel), effective (Baker;
+best bounds Stark, Juricevic); completely tabulated for |k| ≤ 10⁷
+(Gebel–Pethő–Zimmer for 10⁴; Bennett–Ghadermarzi for 10⁷).
+**Software.** Sage: `EllipticCurve([0,0,0,0,k]).integral_points()`; Magma ditto;
+Bennett–Ghadermarzi online tables.
+**References.** Mordell 1913; Gebel–Pethő–Zimmer, "On Mordell's equation" (Compositio 110, 1998);
+Bennett–Ghadermarzi, "Mordell's equation: a classical approach" (LMS JCM 18, 2015).
+
 ### `cubic-surface` — Cubic surfaces / del Pezzo — P3, research
 **Form.** F(x, y, z, w) = 0 cubic (e.g. diagonal ax³+by³+cz³+dw³ = 0).
 **Status.** Rational points conjecturally dense once one exists (unirationality);

tests/test_classify.py

@@ -23,6 +23,9 @@
     ("x^2 + x*y + y^2 + z^2 = 14", "", "quadratic-form-representation"),
     ("x^2 + y^2 - z^2 + 3*x - 7 = 0", "", "quadric"),
     # genus one
+    ("y^2 = x^3 - 2", "", "mordell"),
+    ("y^2 = x^3 + k", "k", "mordell"),
+    ("x^3 = y^2 - 2", "", "mordell"),
     ("y^2 + y = x^3 - x^2 - 10*x - 20", "", "elliptic-weierstrass"),
     # higher-genus curves and binary forms
     ("x^3 + 2*y^3 = 11", "", "thue"),
@@ -60,6 +63,12 @@ def test_reducible():
     assert sorted(c.slug for c in cls.components) == ["univariate", "univariate"]
 
 
+def test_mordell_data():
+    cls = classify("y^2 = x^3 - 2")
+    assert cls.data["k"] == "-2"
+    assert "elliptic-weierstrass" in cls.lineage
+
+
 def test_parametric_quadratic_form_classifies():
     """A parametric quadratic form must classify, not raise (matchers._gram)."""
     cls = classify("x^2 + y^2 = D*z^2", params="D")
@@ -73,6 +82,11 @@ def test_gen_fermat_regimes():
     assert hyperbolic.data["regime"] == "hyperbolic"
 
 
+def test_params_as_list():
+    cls = classify("y^2 = x^3 + k", params=["k"])
+    assert cls.slug == "mordell"
+
+
 def test_match_lookup_by_slug():
     cls = classify("3*x + 5*y = 1")
     assert cls.match_for("linear").slug == "linear"

tests/test_solvers.py

@@ -67,6 +67,13 @@ def test_univariate():
     assert [t[0] for t in s.solutions] == [2, 3]
 
 
+def test_mordell_integral_points():
+    s = solve("y^2 = x^3 - 2")
+    # unknowns are ordered (y, x) by appearance in the input
+    assert set(s.solutions) == {(5, 3), (-5, 3)}
+    assert s.complete
+
+
 def test_thue():
     s = solve("x^3 + 2*y^3 = 11")
     assert (3, -2) in s.solutions
@@ -207,6 +214,13 @@ def capture(cls, match):
     assert "coeffs" not in seen["data"]    # ... not the linear match's
 
 
+def test_curve_role_reversal_is_transported_back():
+    S = solve("u^2 = v^3 - 2")
+    assert S.variables == ("u", "v")
+    assert sorted(S.solutions) == [(-5, 3), (5, 3)]
+    assert_valid_solutions("u^2 = v^3 - 2", S.solutions)
+
+
 def test_component_of_a_rational_equation_keeps_the_parent_conditions():
     """(x - 2)(x - y)/y = 0 factors; y != 0 still binds each component."""
     cls = classify("(x - 2)*(x - y)/y = 0")