KNOTT-6.1 — Fibonacci Generator of Primitive Pythagorean Triples¶
Curated Source & Catalog Provenance¶
Catalog ID |
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Curated Source |
Dr. Ron Knott, Pythagorean Triangles and Triples (#pythfib) |
Source Attribution |
© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition |
Exact SymPy Validator |
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GEOMETOR Concordance |
Constructive Synthesis & Exact Proof¶
On Pythag/pythag.html (KnottPythag), Dr. Ron Knott catalogs
Alwyn Horadam’s (Horadam1961) four-term Fibonacci generator
of Pythagoras’s right triangles: any four consecutive positive
Fibonacci numbers \((F_n, F_{n+1}, F_{n+2}, F_{n+3})\) satisfy
Nested Exact Right Triangles & Rational Incircle: Anchored at right-angle vertex O, \(n = 1 \mapsto (1, 1, 2, 3)\) constructs the classic \((3, 4, 5)\) triangle OA1B1 with hypotenuse A1B1 \(= F_5 = 5\), while \(n = 2 \mapsto (1, 2, 3, 5)\) constructs the \((5, 12, 13)\) triangle OA2B2 on legs OA2 \(= 5\), OB2 \(= 12\), and hypotenuse A2B2 \(= F_7 = 13\). Its exact incircle (I2, T2) centered at I2 \((2, 2)\) has integer inradius \(r = F_2 F_3 = 2\) and triangle area \(F_2 F_3 F_4 F_5 = 1 \cdot 2 \cdot 3 \cdot 5 = 30\).
Divisibility Dichotomy: Because \(\gcd(F_{n+1}, F_{n+2}) = 1\) and \(2 \mid F_k \iff 3 \mid k\), the generated triple has \(\gcd = 1\) (primitive) whenever \(3 \nmid n\) and \(\gcd = 2\) (twice a primitive triple, e.g. \(n = 3 \mapsto (16, 30, 34) = 2(8, 15, 17)\)) whenever \(3 \mid n\).
Machine-Verified Exact Symbolic Invariants¶
Invariant |
Exact Symbolic Value |
|---|---|
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\(5\) |
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\(6\) |
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\(13\) |
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\(30\) |
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\(2\) |
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\(89\) |
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\(1560\) |