diff --git a/diophantine_classifier/data/families/simultaneous-pell.yaml b/diophantine_classifier/data/families/simultaneous-pell.yaml new file mode 100644 index 0000000..693a174 --- /dev/null +++ b/diophantine_classifier/data/families/simultaneous-pell.yaml @@ -0,0 +1,20 @@ +# Family: Simultaneous Pell equations +# 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: simultaneous-pell +name: Simultaneous Pell equations +priority: 2 +status: effective +class: quadratic +form: x^2 - a*z^2 = 1, y^2 - b*z^2 = 1 +parents: +- pell-like +matcher: false +methods: +- linear forms in logarithms +- LLL reduction +references: +- key: Anglin1996 + why: practical algorithm for simultaneous Pell systems +- key: Bennett1998 + why: sharp bound (at most three) on the number of solutions diff --git a/docs/FAMILIES.md b/docs/FAMILIES.md index 86434cf..7ef4408 100644 --- a/docs/FAMILIES.md +++ b/docs/FAMILIES.md @@ -89,6 +89,15 @@ theory. **References.** Lagrange–Matthews–Mollin; Matthews, "The Diophantine equation x²−Dy²=N" (2000); Mollin, *Fundamental Number Theory with Applications*. +### `simultaneous-pell` — Simultaneous Pell equations — P2, effective +**Form.** x² − az² = 1, y² − bz² = 1 (and variants sharing a variable). +**Status.** Finitely many; effective via linear forms in logarithms; at most 3 +solutions in many regimes (Bennett); practical resolution via LLL reduction of the +Baker bound. +**Software.** No turnkey solver; scripts on top of Sage/PARI following Anglin/de Weger. +**References.** Anglin 1996; Bennett, "On the number of solutions of simultaneous +Pell equations" (J. reine angew. Math. 498, 1998). + ### `legendre` — Legendre / diagonal ternary quadratic — P1, algorithmic **Form.** ax² + by² + cz² = 0 (nontrivial solutions; usually abc squarefree, mixed signs). **Status.** Solvability by Legendre's criterion / Hasse–Minkowski; when solvable, a