Family: goormaghtigh — Goormaghtigh equation

diophantine_classifier/data/families/goormaghtigh.yaml

@@ -0,0 +1,16 @@
+# Family: Goormaghtigh 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: goormaghtigh
+name: Goormaghtigh equation
+priority: 3
+status: open
+class: expdioph
+form: (x^m - 1)/(x - 1) = (y^n - 1)/(y - 1)
+parents:
+- exponential-diophantine
+matcher: false
+notes: 'Known: 31 and 8191; finite effective for fixed bases (Balasubramanian-Shorey).'
+references:
+- key: BalasubramanianShorey1980
+  why: effective finiteness for fixed bases

docs/FAMILIES.md

@@ -267,6 +267,12 @@ side of the decidability boundary.
 **References.** Shorey–Tijdeman 1986; Evertse–Schlickewei–Schmidt 2002 (subspace);
 Davis–Putnam–Robinson 1961.
 
+### `goormaghtigh` — Goormaghtigh equation — P3, open
+**Form.** (xᵐ − 1)/(x − 1) = (yⁿ − 1)/(y − 1), y > x ≥ 2, m > n ≥ 3.
+**Status.** Known: 31 = (2⁵−1)/1 = (5³−1)/4 and 8191 (repunits in bases 2/90);
+conjecturally all; finite effective for fixed (x, y) (Balasubramanian–Shorey 1980).
+**References.** Goormaghtigh 1917; Balasubramanian–Shorey 1980.
+
 
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