Family: ramanujan-nagell — Generalized Ramanujan-Nagell equation

diophantine_classifier/data/families/ramanujan-nagell.yaml

@@ -0,0 +1,26 @@
+# Family: Generalized Ramanujan-Nagell 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: ramanujan-nagell
+name: Generalized Ramanujan-Nagell equation
+priority: 1
+status: effective
+class: expdioph
+form: 'x^2 + d = k*b^n (classical: x^2 + 7 = 2^n)'
+parents:
+- polynomial-exponential
+matcher: true
+methods:
+- hypergeometric method (Beukers)
+- Baker + LLL (Petho-de Weger)
+notes: 'Classical case: n in {3,4,5,7,15}. At most 2 solutions for general d (Beukers).'
+examples:
+- x^2 + 7 = 2^n
+- x^2 + 11 = 3^n
+references:
+- key: Nagell1961
+  why: the proof that n lies in {3, 4, 5, 7, 15}
+- key: Beukers1981
+  why: 'sharp bounds: at most two solutions of the generalized equation'
+- key: deWeger1989
+  why: the Baker + LLL toolkit for concrete instances

diophantine_classifier/solvers.py

@@ -1154,6 +1154,30 @@ def rec(k_left, target, minimum, acc):
         "permutations give the rest", complete=True)
 
 
+def _solve_ramanujan_nagell(cls, match):
+    r"""
+    The classical Ramanujan-Nagell equation ``x^2 + 7 = 2^n``.
+
+    EXAMPLES::
+
+        sage: from diophantine_classifier import solve
+        sage: solve("x^2 + 7 = 2^n").solutions
+        [(1, 3), (3, 4), (5, 5), (11, 7), (181, 15)]
+    """
+    d, k, base = _zz(match.data, "d"), _zz(match.data, "k"), _zz(match.data, "base")
+    if (d, k, base) != (7, 1, 2):
+        raise SolverUnavailable(
+            "only the classical x^2 + 7 = 2^n is hardwired; general (d, k, b) "
+            "need a Baker + LLL computation (Petho-de Weger)")
+    pairs = [(ZZ(x), ZZ(n))
+             for x, n in [(1, 3), (3, 4), (5, 5), (11, 7), (181, 15)]]
+    return SolutionSet(
+        _normalized(match), pairs, "finite-complete",
+        "Nagell's theorem: n ∈ {3, 4, 5, 7, 15} (x > 0 shown; -x symmetric)",
+        complete=True)
+
+
+#: dispatch table: family slug -> solver function
 SOLVERS = {
     "univariate": _solve_univariate,
     "linear": _solve_linear,
@@ -1164,6 +1188,7 @@ def rec(k_left, target, minimum, acc):
     "elliptic-weierstrass": _solve_weierstrass,
     "thue": _solve_thue,
     "egyptian-fractions": _solve_egyptian,
+    "ramanujan-nagell": _solve_ramanujan_nagell,
 }
 
 

docs/FAMILIES.md

@@ -248,6 +248,17 @@ equations: a miscellany" (Int. J. Number Theory 11, 2015).
 
 ## 6. Polynomial–exponential equations
 
+### `ramanujan-nagell` — (Generalized) Ramanujan–Nagell — P1, effective
+**Form.** x² + d = k·bⁿ (classical: x² + 7 = 2ⁿ); more generally f(x) = k·bⁿ with
+f quadratic.
+**Status.** Classical case: exactly n ∈ {3,4,5,7,15} (conjectured Ramanujan 1913,
+proved Nagell 1948). Generalized: at most 2 solutions apart from finitely many
+explicit exceptional d (Apéry 1960, Beukers 1981 with sharp bounds — hypergeometric
+method); fully effective; practical resolution via Baker + LLL (Pethő–de Weger).
+**Software.** scripts via PARI/Sage (no standard intrinsic); de Weger's algorithms.
+**References.** Ramanujan 1913; Nagell 1948; Apéry 1960; Beukers 1981;
+de Weger, *Algorithms for Diophantine Equations* (1989).
+
 ### `power-values` — Power values of polynomials (Schinzel–Tijdeman) — P2, effective in n
 **Form.** f(x) = c·yⁿ, f fixed polynomial with ≥ 2 distinct roots, n ≥ 2 unknown.
 **Status.** n is effectively bounded (Schinzel–Tijdeman 1976); for each fixed n it

tests/test_classify.py

@@ -39,6 +39,8 @@
     # diagonal / surfaces
     ("x^4 + y^4 + z^4 = w^4", "", "equal-sums-like-powers"),   # Elkies
     # polynomial-exponential
+    ("x^2 + 7 = 2^n", "", "ramanujan-nagell"),
+    ("x^2 + 11 = 3^n", "", "ramanujan-nagell"),
     ("x^3 - 4 = y^n", "", "power-values"),
     # unit fractions
     ("1/x + 1/y + 1/z = 1", "", "egyptian-fractions"),
@@ -73,6 +75,12 @@ def test_gen_fermat_regimes():
     assert hyperbolic.data["regime"] == "hyperbolic"
 
 
+def test_explain_smoke():
+    text = classify("x^2 + 7 = 2^n").explain()
+    assert "ramanujan-nagell" in text
+    assert "Nagell" in text or "1948" in text
+
+
 def test_match_lookup_by_slug():
     cls = classify("3*x + 5*y = 1")
     assert cls.match_for("linear").slug == "linear"

tests/test_solvers.py

@@ -74,6 +74,12 @@ def test_thue():
     assert_valid_solutions("x^3 + 2*y^3 = 11", s.solutions)
 
 
+def test_ramanujan_nagell():
+    s = solve("x^2 + 7 = 2^n")
+    assert (11, 7) in s.solutions and len(s.solutions) == 5
+    assert s.complete
+
+
 def test_unavailable_carries_hints():
     with pytest.raises(SolverUnavailable) as err:
         solve("y^2 = x^7 + 3")